Mathematical relations qrsFrom Wikipedia, the free encyclopedia

Contents

1 Process calculus 11.1 Essential features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Mathematics of processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Parallel composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Sequential composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.4 Reduction semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.5 Hiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.6 Recursion and replication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.7 Null process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Discrete and continuous process algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.5 Current research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Software implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Relationship to other models of concurrency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Quasitransitive relation 62.1 Formal definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Quotient by an equivalence relation 83.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Rational consequence relation 10

i

ii CONTENTS

4.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Uses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.3 Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.4 Rational consequence relations via atom preferences . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.4.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.5 The representation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.5.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Reduct 135.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

6 Reflexive closure 146.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

7 Reflexive relation 157.1 Related terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157.3 Number of reflexive relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.4 Philosophical logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

8 Relation algebra 198.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8.1.1 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 Expressing properties of binary relations in RA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208.3 Expressive power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

8.3.1 Q-Relation Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218.5 Historical remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.8 Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

CONTENTS iii

8.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

9 Relation construction 249.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

10 Representation (mathematics) 2510.1 Representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

10.2.1 Graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2.2 Order theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2510.2.3 Polysemy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

10.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

11 Separoid 2811.1 The axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2811.3 The basic lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2911.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

12 Series-parallel partial order 3012.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.2 Forbidden suborder characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.3 Order dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3112.4 Connections to graph theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.5 Computational complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3212.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3312.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

13 Surjective function 3413.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3513.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

13.3.1 Surjections as right invertible functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3613.3.2 Surjections as epimorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3713.3.3 Surjections as binary relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.4 Cardinality of the domain of a surjection . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.5 Composition and decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.3.6 Induced surjection and induced bijection . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

13.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3813.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3913.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

iv CONTENTS

14 Symmetric closure 4014.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.2 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

15 Symmetric relation 4115.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

15.1.1 In mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4115.1.2 Outside mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

15.2 Relationship to asymmetric and antisymmetric relations . . . . . . . . . . . . . . . . . . . . . . . 4215.3 Additional aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4215.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4315.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 44

15.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4415.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4515.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Chapter 1

Process calculus

In computer science, the process calculi (or process algebras) are a diverse family of related approaches for for-mally modelling concurrent systems. Process calculi provide a tool for the high-level description of interactions,communications, and synchronizations between a collection of independent agents or processes. They also providealgebraic laws that allow process descriptions to be manipulated and analyzed, and permit formal reasoning aboutequivalences between processes (e.g., using bisimulation). Leading examples of process calculi include CSP, CCS,ACP, and LOTOS.[1] More recent additions to the family include the π-calculus, the ambient calculus, PEPA, thefusion calculus and the join-calculus.

1.1 Essential features

While the variety of existing process calculi is very large (including variants that incorporate stochastic behaviour,timing information, and specializations for studying molecular interactions), there are several features that all processcalculi have in common:[2]

• Representing interactions between independent processes as communication (message-passing), rather than asmodification of shared variables.

• Describing processes and systems using a small collection of primitives, and operators for combining thoseprimitives.

• Defining algebraic laws for the process operators, which allow process expressions to be manipulated usingequational reasoning.

1.2 Mathematics of processes

To define a process calculus, one starts with a set of names (or channels) whose purpose is to provide means ofcommunication. In many implementations, channels have rich internal structure to improve efficiency, but this isabstracted away in most theoretic models. In addition to names, one needs a means to form new processes from old.The basic operators, always present in some form or other, allow:[3]

• parallel composition of processes

• specification of which channels to use for sending and receiving data

• sequentialization of interactions

• hiding of interaction points

• recursion or process replication

1

2 CHAPTER 1. PROCESS CALCULUS

1.2.1 Parallel composition

Parallel composition of two processes P and Q , usually written P |Q , is the key primitive distinguishing the processcalculi from sequential models of computation. Parallel composition allows computation in P and Q to proceedsimultaneously and independently. But it also allows interaction, that is synchronisation and flow of information fromP to Q (or vice versa) on a channel shared by both. Crucially, an agent or process can be connected to more than onechannel at a time.Channels may be synchronous or asynchronous. In the case of a synchronous channel, the agent sending a messagewaits until another agent has received the message. Asynchronous channels do not require any such synchronization.In some process calculi (notably the π-calculus) channels themselves can be sent in messages through (other) channels,allowing the topology of process interconnections to change. Some process calculi also allow channels to be createdduring the execution of a computation.

1.2.2 Communication

Interaction can be (but isn't always) a directed flow of information. That is, input and output can be distinguished asdual interaction primitives. Process calculi that make such distinctions typically define an input operator (e.g. x(v)) and an output operator (e.g. x⟨y⟩ ), both of which name an interaction point (here x ) that is used to synchronisewith a dual interaction primitive.Information should be exchanged, it will flow from the outputting to the inputting process. The output primitive willspecify the data to be sent. In x⟨y⟩ , this data is y . Similarly, if an input expects to receive data, one or more boundvariables will act as place-holders to be substituted by data, when it arrives. In x(v) , v plays that role. The choice ofthe kind of data that can be exchanged in an interaction is one of the key features that distinguishes different processcalculi.

1.2.3 Sequential composition

Sometimes interactions must be temporally ordered. For example, it might be desirable to specify algorithms such as:first receive some data on x and then send that data on y . Sequential composition can be used for such purposes. It iswell known from other models of computation. In process calculi, the sequentialisation operator is usually integratedwith input or output, or both. For example, the process x(v) · P will wait for an input on x . Only when this inputhas occurred will the process P be activated, with the received data through x substituted for identifier v .

1.2.4 Reduction semantics

The key operational reduction rule, containing the computational essence of process calculi, can be given solely interms of parallel composition, sequentialization, input, and output. The details of this reduction vary among thecalculi, but the essence remains roughly the same. The reduction rule is:

x⟨y⟩ · P | x(v) ·Q −→ P | Q[y/v]

The interpretation of this reduction rule is:

1. The process x⟨y⟩ ·P sends a message, here y , along the channel x . Dually, the process x(v) ·Q receives thatmessage on channel x .

2. Once the message has been sent, x⟨y⟩ · P becomes the process P , while x(v) ·Q becomes the process Q[y/v], which is Q with the place-holder v substituted by y , the data received on x .

The class of processes that P is allowed to range over as the continuation of the output operation substantially influ-ences the properties of the calculus.

1.3. DISCRETE AND CONTINUOUS PROCESS ALGEBRA 3

1.2.5 Hiding

Processes do not limit the number of connections that can be made at a given interaction point. But interaction pointsallow interference (i.e. interaction). For the synthesis of compact, minimal and compositional systems, the ability torestrict interference is crucial. Hiding operations allow control of the connections made between interaction pointswhen composing agents in parallel. Hiding can be denoted in a variety of ways. For example, in the π -calculus thehiding of a name x in P can be expressed as (ν x)P , while in CSP it might be written as P \ {x} .

1.2.6 Recursion and replication

The operations presented so far describe only finite interaction and are consequently insufficient for full computability,which includes non-terminating behaviour. Recursion and replication are operations that allow finite descriptionsof infinite behaviour. Recursion is well known from the sequential world. Replication !P can be understood asabbreviating the parallel composition of a countably infinite number of P processes:

!P = P |!P

1.2.7 Null process

Process calculi generally also include a null process (variously denoted as nil , 0 , STOP , δ , or some other appropriatesymbol) which has no interaction points. It is utterly inactive and its sole purpose is to act as the inductive anchor ontop of which more interesting processes can be generated.

1.3 Discrete and continuous process algebra

Process algebra has been studied for discrete time and continuous time (real time or dense time).[4]

1.4 History

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a com-putable function, with μ-recursive functions, Turing Machines and the lambda calculus possibly being the best-knownexamples today. The surprising fact that they are essentially equivalent, in the sense that they are all encodable intoeach other, supports the Church-Turing thesis. Another shared feature is more rarely commented on: they all aremost readily understood as models of sequential computation. The subsequent consolidation of computer science re-quired a more subtle formulation of the notion of computation, in particular explicit representations of concurrencyand communication. Models of concurrency such as the process calculi, Petri nets in 1962, and the Actor model in1973 emerged from this line of enquiry.Research on process calculi began in earnest with Robin Milner's seminal work on the Calculus of CommunicatingSystems (CCS) during the period from 1973 to 1980. C.A.R. Hoare's Communicating Sequential Processes (CSP)first appeared in 1978, and was subsequently developed into a full-fledged process calculus during the early 1980s.There was much cross-fertilization of ideas between CCS and CSP as they developed. In 1982 Jan Bergstra andJan Willem Klop began work on what came to be known as the Algebra of Communicating Processes (ACP), andintroduced the term process algebra to describe their work.[1] CCS, CSP, and ACP constitute the three major branchesof the process calculi family: the majority of the other process calculi can trace their roots to one of these three calculi.

1.5 Current research

Various process calculi have been studied and not all of them fit the paradigm sketched here. The most prominentexample may be the ambient calculus. This is to be expected as process calculi are an active field of study. Currentlyresearch on process calculi focuses on the following problems.

4 CHAPTER 1. PROCESS CALCULUS

• Developing new process calculi for better modeling of computational phenomena.

• Finding well-behaved subcalculi of a given process calculus. This is valuable because (1) most calculi are fairlywild in the sense that they are rather general and not much can be said about arbitrary processes; and (2)computational applications rarely exhaust the whole of a calculus. Rather they use only processes that are veryconstrained in form. Constraining the shape of processes is mostly studied by way of type systems.

• Logics for processes that allow one to reason about (essentially) arbitrary properties of processes, following theideas of Hoare logic.

• Behavioural theory: what does it mean for two processes to be the same? How can we decide whether twoprocesses are different or not? Can we find representatives for equivalence classes of processes? Generally,processes are considered to be the same if no context, that is other processes running in parallel, can detect adifference. Unfortunately, making this intuition precise is subtle and mostly yields unwieldy characterisations ofequality (which in most cases must also be undecidable, as a consequence of the halting problem). Bisimulationsare a technical tool that aids reasoning about process equivalences.

• Expressivity of calculi. Programming experience shows that certain problems are easier to solve in somelanguages than in others. This phenomenon calls for a more precise characterisation of the expressivity ofcalculi modeling computation than that afforded by the Church-Turing thesis. One way of doing this is toconsider encodings between two formalisms and see what properties encodings can potentially preserve. Themore properties can be preserved, the more expressive the target of the encoding is said to be. For processcalculi, the celebrated results are that the synchronous π -calculus is more expressive than its asynchronousvariant, has the same expressive power as the higher-order π -calculus, but is less than the ambient calculus.

• Using process calculus to model biological systems (stochasticπ -calculus, BioAmbients, Beta Binders, BioPEPA,Brane calculus). It is thought by some that the compositionality offered by process-theoretic tools can help bi-ologists to organise their knowledge more formally.

1.6 Software implementations

The ideas behind process algebra have given rise to several tools including:

• CADP

• Concurrency Workbench

• mCRL2 toolset

1.7 Relationship to other models of concurrency

The history monoid is the free object that is generically able to represent the histories of individual communicatingprocesses. A process calculus is then a formal language imposed on a history monoid in a consistent fashion.[5] Thatis, a history monoid can only record a sequence of events, with synchronization, but does not specify the allowedstate transitions. Thus, a process calculus is to a history monoid what a formal language is to a free monoid (a formallanguage is a subset of the set of all possible finite-length strings of an alphabet generated by the Kleene star).The use of channels for communication is one of the features distinguishing the process calculi from other models ofconcurrency, such as Petri nets and the Actor model (see Actor model and process calculi). One of the fundamentalmotivations for including channels in the process calculi was to enable certain algebraic techniques, thereby makingit easier to reason about processes algebraically.

1.8 See also• Stochastic probe

1.9. REFERENCES 5

1.9 References[1] Baeten, J.C.M. (2004). “A brief history of process algebra” (PDF). Rapport CSR 04-02 (Vakgroep Informatica, Technische

Universiteit Eindhoven).

[2] Pierce, Benjamin. “Foundational Calculi for Programming Languages”. The Computer Science and Engineering Handbook.CRC Press. pp. 2190–2207. ISBN 0-8493-2909-4.

[3] Baeten, J.C.M.; Bravetti, M. (August 2005). “A Generic Process Algebra”. Algebraic Process Calculi: The First TwentyFive Years and Beyond (BRICS Notes Series NS-05-3). Bertinoro, Forl`ı, Italy: BRICS, Department of Computer Science,University of Aarhus. Retrieved 2007-12-29.

[4] Baeten, J. C. M.; Middelburg, C. A. “Process algebra with timing: Real time and discrete time”. CiteSeerX: 10 .1 .1 .42 .729.

[5] Mazurkiewicz, Antoni (1995). “Introduction to Trace Theory”. In Diekert, V.; Rozenberg, G. The Book of Traces(POSTSCRIPT). Singapore: World Scientific. pp. 3–41. ISBN 981-02-2058-8.

1.10 Further reading• Matthew Hennessy: Algebraic Theory of Processes, The MIT Press, ISBN 0-262-08171-7.

• C. A. R. Hoare: Communicating Sequential Processes, Prentice Hall, ISBN 0-13-153289-8.

• This book has been updated by Jim Davies at the Oxford University Computing Laboratory and the newedition is available for download as a PDF file at the Using CSP website.

• Robin Milner: A Calculus of Communicating Systems, Springer Verlag, ISBN 0-387-10235-3.

• Robin Milner: Communicating and Mobile Systems: the Pi-Calculus, Springer Verlag, ISBN 0-521-65869-1.

• Andrew Mironov: Theory of processes

Chapter 2

Quasitransitive relation

Quasitransitivity is a weakened version of transitivity that is used in social choice theory or microeconomics. In-formally, a relation is quasitransitive if it is symmetric for some values and transitive elsewhere. The concept wasintroduced by Sen (1969) to study the consequences of Arrow’s theorem.

2.1 Formal definition

A binary relation T over a set X is quasitransitive if for all a, b, and c in X the following holds:

(aT b) ∧ ¬(bT a) ∧ (bT c) ∧ ¬(cT b) ⇒ (aT c) ∧ ¬(cT a).

If the relation is also antisymmetric, T is transitive.Alternately, for a relation T, define the asymmetric or “strict” part P:

(aP b) ⇔ (aT b) ∧ ¬(bT a).

Then T is quasitransitive iff P is transitive.

2.2 Examples

Preferences are assumed to be quasitransitive (rather than transitive) in some economic contexts. The classic exampleis a person indifferent between 10 and 11 grams of sugar and indifferent between 11 and 12 grams of sugar, but whoprefers 12 grams of sugar to 10. Similarly, the Sorites paradox can be resolved by weakening assumed transitivity ofcertain relations to quasitransitivity.

2.3 Properties

• Every transitive relation is quasitransitive; every quasitransitive relation is an acyclic relation. In each case theconverse does not hold in general.

2.4 See also

• Intransitivity

• Reflexive relation

6

2.5. REFERENCES 7

2.5 References• Bossert, Walter; Suzumura, Kōtarō (2010). Consistency, choice and rationality. Harvard University Press.

ISBN 0674052994.

• Sen, A. (1969). “Quasi-transitivity, rational choice and collective decisions”. Rev. Econ. Stud. 36: 381–393.doi:10.2307/2296434. Zbl 0181.47302.

Chapter 3

Quotient by an equivalence relation

This article is about a generalization to category theory, used in scheme theory. For the common meaning, seeEquivalence class.

In mathematics, given a category C, a quotient of an object X by an equivalence relation f : R → X × X is acoequalizer for the pair of maps

Rf→X ×X

pri→X, i = 1, 2,

where R is an object in C and "f is an equivalence relation” means that, for any object T in C, the image (which isa set) of f : R(T ) = Mor(T,R) → X(T ) × X(T ) is an equivalence relation; that is, (x, y) is in it if and only if(y, x) is in it, etc.The basic case in practice is when C is the category of all schemes over some scheme S. But the notion is flexible andone can also take C to be the category of sheaves.

3.1 Examples

• Let X be a set and consider some equivalence relation on it. Let Q be the set of all equivalence classes in X.Then the map q : X → Q that sends an element x to an equivalence class to which x belong is a quotient.

• In the above example, Q is a subset of the power set H of X. In algebraic geometry, one might replace Hby a Hilbert scheme or disjoint union of Hilbert schemes. In fact, Grothendieck constructed a relative Picardscheme of a flat projective schemeX[1] as a quotientQ (of the scheme Z parametrizing relative effective divisorson X) that is a closed scheme of a Hilbert scheme H. The quotient map q : Z → Q can then be thought of asa relative version of the Abel map.

3.2 See also

• categorical quotient, a special case

3.3 Notes

[1] One also needs to assume the geometric fibers are integral schemes; Mumford’s example shows the “integral” cannot beomitted.

8

3.4. REFERENCES 9

3.4 References• Nitsure, N. Construction of Hilbert and Quot schemes. Fundamental algebraic geometry: Grothendieck’s FGA

explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137.

Chapter 4

Rational consequence relation

In logic, a rational consequence relation is a non-monotonic consequence relation satisfying certain properties listedbelow.

4.1 Properties

A rational consequence relation satisfies:

REF Reflexivity θ ⊢ θ

and the so-called Gabbay-Makinson rules:

LLE Left Logical Equivalence θ⊢ψ θ≡ϕϕ⊢ψ

RWE Right-hand weakening θ⊢ϕ ϕ|=ψθ⊢ψ

CMO Cautious monotonicity θ⊢ϕ θ⊢ψθ∧ψ⊢ϕ

DIS Logical or (ie disjunction) on left hand side θ⊢ψ ϕ⊢ψθ∨ϕ⊢ψ

AND Logical and on right hand side θ⊢ϕ θ⊢ψθ⊢ϕ∧ψ

RMO Rational monotonicity ϕ̸⊢¬θ ϕ⊢ψϕ∧θ⊢ψ

4.2 Uses

The rational consequence relation is non-monotonic, and the relation θ ⊢ ϕ is intended to carry the meaning thetausually implies phi or phi usually follows from theta. In this sense it is more useful for modeling some everydaysituations than a monotone consequence relation because the latter relation models facts in a more strict booleanfashion - something either follows under all circumstances or it does not.

4.2.1 Example

The statement “If a cake contains sugar then it tastes good” implies under a monotone consequence relation the state-ment “If a cake contains sugar and soap then it tastes good.” Clearly this doesn't match our own understanding ofcakes. By asserting “If a cake contains sugar then it usually tastes good” a rational consequence relation allows fora more realistic model of the real world, and certainly it does not automatically follow that “If a cake contains sugarand soap then it usually tastes good.”

Note that if we also have the information “If a cake contains sugar then it usually contains butter” then we may legallyconclude (under CMO) that “If a cake contains sugar and butter then it usually tastes good.”. Equally in the absence

10

4.3. CONSEQUENCES 11

of a statement such as “If a cake contains sugar then usually it contains no soap" then we may legally conclude fromRMO that “If the cake contains sugar and soap then it usually tastes good.”If this latter conclusion seems ridiculous to you then it is likely that you are subconsciously asserting your own pre-conceived knowledge about cakes when evaluating the validity of the statement. That is, from your experience youknow that cakes which contain soap are likely to taste bad so you add to the system your own knowledge such as“Cakes which contain sugar do not usually contain soap.”, even though this knowledge is absent from it. If the conclu-sion seems silly to you then you might consider replacing the word soap with the word eggs to see if it changes yourfeelings.

4.2.2 Example

Consider the sentences:

• Young people are usually happy

• Drug abusers are usually not happy

• Drug abusers are usually young

We may consider it reasonable to conclude:

• Young drug abusers are usually not happy

This would not be a valid conclusion under a monotonic deduction system (omitting of course the word 'usually'),since the third sentence would contradict the first two. In contrast the conclusion follows immediately using theGabbay-Makinson rules: applying the rule CMO to the last two sentences yields the result.

4.3 Consequences

The following consequences follow from the above rules:

MP Modus ponens θ⊢ϕ θ⊢(ϕ→ψ)θ⊢ψ

MP is proved via the rules AND and RWE.

CON Conditionalisation θ∧ϕ⊢ψθ⊢(ϕ→ψ)

CC Cautious Cut θ⊢ϕ θ∧ϕ⊢ψθ⊢ψ

The notion of Cautious Cut simply encapsulates the operation of conditionalisation, followed by MP.It may seem redundant in this sense, but it is often used in proofs so it is useful to have a name forit to act as a shortcut.

SCL Supraclassity θ|=ϕθ⊢ϕ

SCL is proved trivially via REF and RWE.

4.4 Rational consequence relations via atom preferences

Let L = {p1, . . . , pn} be a finite language. An atom is a formula of the form∧ni=1 p

ϵi (where p1 = p and p−1 = ¬p

). Notice that there is a unique valuation which makes any given atom true (and conversely each valuation satisfiesprecisely one atom). Thus an atom can be used to represent a preference about what we believe ought to be true.Let AtL be the set of all atoms in L. For θ ∈ SL, define Sθ = {α ∈ AtL|α |=SC θ} .Let s⃗ = s1, . . . , sm be a sequence of subsets of AtL . For θ , ϕ in SL, let the relation ⊢s⃗ be such that θ ⊢s⃗ ϕ if oneof the following holds:

12 CHAPTER 4. RATIONAL CONSEQUENCE RELATION

1. Sθ ∩ si = ∅ for each 1 ≤ i ≤ m

2. Sθ ∩ si ̸= ∅ for some 1 ≤ i ≤ m and for the least such i, Sθ ∩ si ⊆ Sϕ .

Then the relation ⊢s⃗ is a rational consequence relation. This may easily be verified by checking directly that it satisfiesthe GM-conditions.The idea behind the sequence of atom sets is that the earlier sets account for the most likely situations such as “youngpeople are usually law abiding” whereas the later sets account for the less likely situations such as “young joyridersare usually not law abiding”.

4.4.1 Notes

1. By the definition of the relation ⊢s⃗ , the relation is unchanged if we replace s2 with s2 \s1 , s3 with s3 \s2 \s1... and sm with sm \

∪m−1i=1 si . In this way we make each si disjoint. Conversely it makes no difference to the

rcr ⊢s⃗ if we add to subsequent si atoms from any of the preceding si .

4.5 The representation theorem

It can be proven that any rational consequence relation on a finite language is representable via a sequence of atompreferences above. That is, for any such rational consequence relation ⊢ there is a sequence s⃗ = s1, . . . , sm of subsetsof AtL such that the associated rcr ⊢s⃗ is the same relation: ⊢s⃗=⊢

4.5.1 Notes

1. By the above property of ⊢s⃗ , the representation of an rcr ⊢ need not be unique - if the si are not disjoint thenthey can be made so without changing the rcr and conversely if they are disjoint then each subsequent set cancontain any of the atoms of the previous sets without changing the rcr.

4.6 References• A mathematical paper in which the GM rules are defined

Chapter 5

Reduct

This article is about a relation on algebraic structures. For reducts in abstract rewriting, see Confluence (abstractrewriting).

In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of theoperations and relations of that structure. The converse of “reduct” is “expansion.”

5.1 Definition

Let A be an algebraic structure (in the sense of universal algebra) or equivalently a structure in the sense of modeltheory, organized as a set X together with an indexed family of operations and relations φᵢ on that set, with index setI. Then the reduct of A defined by a subset J of I is the structure consisting of the set X and J-indexed family ofoperations and relations whose j-th operation or relation for j∈J is the j-th operation or relation of A. That is, thisreduct is the structure A with the omission of those operations and relations φi for which i is not in J.A structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.

5.2 Examples

The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition andnegation, obtained by omitting negation. By contrast, the monoid (N,+,0) of natural numbers under addition is notthe reduct of any group.Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.

5.3 References• Burris, Stanley N.; H. P. Sankappanavar (1981). ACourse in Universal Algebra. Springer. ISBN 3-540-90578-

2.

• Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3.

13

Chapter 6

Reflexive closure

In mathematics, the reflexive closure of a binary relation R on a set X is the smallest reflexive relation on X thatcontains R.For example, if X is a set of distinct numbers and x R y means "x is less than y", then the reflexive closure of R is therelation "x is less than or equal to y".

6.1 Definition

The reflexive closure S of a relation R on a set X is given by

S = R ∪ {(x, x) : x ∈ X}

In words, the reflexive closure of R is the union of R with the identity relation on X.

6.2 See also• Transitive closure

• Symmetric closure

6.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

14

Chapter 7

Reflexive relation

In mathematics, a reflexive relation is a binary relation on a set for which every element is related to itself. In otherwords, a relation ~ on a set S is reflexive when x ~ x holds true for every x in S, formally: when ∀x∈S: x~x holds.[1][2]

An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number isequal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity.

7.1 Related terms

A relation that is irreflexive, or anti-reflexive, is a binary relation on a set where no element is related to itself. Anexample is the “greater than” relation (x>y) on the real numbers. Note that not every relation which is not reflexiveis irreflexive; it is possible to define relations where some elements are related to themselves but others are not (i.e.,neither all nor none are). For example, the binary relation “the product of x and y is even” is reflexive on the set ofeven numbers, irreflexive on the set of odd numbers, and neither reflexive nor irreflexive on the set of natural numbers.A relation ~ on a set S is called quasi-reflexive if every element that is related to some element is also related to itself,formally: if ∀x,y∈S: x~y ⇒ x~x ∧ y~y. An example is the relation “has the same limit as” on the set of sequences ofreal numbers: not every sequence has a limit, and thus the relation is not reflexive, but if a sequence has the samelimit as some sequence, then it has the same limit as itself.The reflexive closure ≃ of a binary relation ~ on a set S is the smallest reflexive relation on S that is a superset of ~.Equivalently, it is the union of ~ and the identity relation on S, formally: (≃) = (~) ∪ (=). For example, the reflexiveclosure of x<y is x≤y.The reflexive reduction, or irreflexive kernel, of a binary relation ~ on a set S is the smallest relation ≆ such that ≆shares the same reflexive closure as ~. It can be seen in a way as the opposite of the reflexive closure. It is equivalentto the complement of the identity relation on S with regard to ~, formally: (≆) = (~) \ (=). That is, it is equivalent to~ except for where x~x is true. For example, the reflexive reduction of x≤y is x<y.

7.2 Examples

Examples of reflexive relations include:

• “is equal to” (equality)

• “is a subset of” (set inclusion)

• “divides” (divisibility)

• “is greater than or equal to”

• “is less than or equal to”

Examples of irreflexive relations include:

15

16 CHAPTER 7. REFLEXIVE RELATION

• “is not equal to”

• “is coprime to” (for the integers>1, since 1 is coprime to itself)

• “is a proper subset of”

• “is greater than”

• “is less than”

7.3 Number of reflexive relations

The number of reflexive relations on an n-element set is 2n2−n.[3]

7.4. PHILOSOPHICAL LOGIC 17

7.4 Philosophical logic

Authors in philosophical logic often use deviating designations. A reflexive and a quasi-reflexive relation in themathematical sense is called a totally reflexive and a reflexive relation in philosophical logic sense, respectively.[4][5]

7.5 See also

• Binary relation

• Symmetric relation

• Transitive relation

18 CHAPTER 7. REFLEXIVE RELATION

• Coreflexive relation

7.6 Notes[1] Levy 1979:74

[2] Relational Mathematics, 2010

[3] On-Line Encyclopedia of Integer Sequences A053763

[4] Alan Hausman, Howard Kahane, Paul Tidman (2013). Logic and Philosophy—AModern Introduction. Wadsworth. ISBN1-133-05000-X. Here: p.327-328

[5] D.S. Clarke, Richard Behling (1998). Deductive Logic — An Introduction to Evaluation Techniques and Logical Theory.University Press of America. ISBN 0-7618-0922-8. Here: p.187

7.7 References• Levy, A. (1979) Basic Set Theory, Perspectives in Mathematical Logic, Springer-Verlag. Reprinted 2002,

Dover. ISBN 0-486-42079-5

• Lidl, R. and Pilz, G. (1998). Applied abstract algebra, Undergraduate Texts in Mathematics, Springer-Verlag.ISBN 0-387-98290-6

• Quine, W. V. (1951). Mathematical Logic, Revised Edition. Reprinted 2003, Harvard University Press. ISBN0-674-55451-5

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press, ISBN 978-0-521-76268-7.

7.8 External links• Hazewinkel, Michiel, ed. (2001), “Reflexivity”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-

010-4

Chapter 8

Relation algebra

Not to be confused with relational algebra, a framework for finitary relations and relational databases.

In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involutioncalled converse, a unary operation. The motivating example of a relation algebra is the algebra 2X² of all binaryrelations on a set X, that is, subsets of the cartesian square X2, with R•S interpreted as the usual composition ofbinary relations R and S, and with the converse of R interpreted as the inverse relation.Relation algebra emerged in the 19th-century work of Augustus De Morgan and Charles Peirce, which culminatedin the algebraic logic of Ernst Schröder. The equational form of relation algebra treated here was developed byAlfred Tarski and his students, starting in the 1940s. Tarski and Givant (1987) applied relation algebra to a variable-free treatment of axiomatic set theory, with the implication that mathematics founded on set theory could itself beconducted without variables.

8.1 Definition

A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘ ) is an algebraic structure equipped with the Boolean operations ofconjunction x∧y, disjunction x∨y, and negation x−, the Boolean constants 0 and 1, the relational operations of com-position x•y and converse x ˘ , and the relational constant I, such that these operations and constants satisfy certainequations constituting an axiomatization of relation algebras. A relation algebra is to a system of binary relationson a set containing the empty (0), complete (1), and identity (I) relations and closed under these five operations as agroup is to a system of permutations of a set containing the identity permutation and closed under composition andinverse.Following Jónsson and Tsinakis (1993) it is convenient to define additional operations x◁y = x•y˘ , and, dually, x▷y= x ˘ •y . Jónsson and Tsinakis showed that I◁x = x▷I, and that both were equal to x ˘ . Hence a relation algebra canequally well be defined as an algebraic structure (L, ∧, ∨, −, 0, 1, •, I, ◁, ▷). The advantage of this signature overthe usual one is that a relation algebra can then be defined in full simply as a residuated Boolean algebra for whichI◁x is an involution, that is, I◁(I◁x) = x . The latter condition can be thought of as the relational counterpart of theequation 1/(1/x) = x for ordinary arithmetic reciprocal, and some authors use reciprocal as a synonym for converse.Since residuated Boolean algebras are axiomatized with finitely many identities, so are relation algebras. Hence thelatter form a variety, the variety RA of relation algebras. Expanding the above definition as equations yields thefollowing finite axiomatization.

8.1.1 Axioms

The axioms B1-B10 below are adapted from Givant (2006: 283), and were first set out by Tarski in 1948.[1]

L is a Boolean algebra under binary disjunction, ∨, and unary complementation ()−:

B1: A ∨ B = B ∨ A

B2: A ∨ (B ∨ C) = (A ∨ B) ∨ C

19

20 CHAPTER 8. RELATION ALGEBRA

B3: (A− ∨ B)− ∨ (A− ∨ B−)− = A

This axiomatization of Boolean algebra is due to Huntington (1933). Note that the meet of the implied Booleanalgebra is not the • operator (even though it distributes over ∨ like a meet does), nor is the 1 of the Boolean algebrathe I constant.L is a monoid under binary composition (•) and nullary identity I:

B4: A•(B•C) = (A•B)•CB5: A•I = A

Unary converse () ˘ is an involution with respect to composition:

B6: A ˘̆ = A

B7: (A•B) ˘ = B ˘ •A ˘

Axiom B6 defines conversion as an involution, whereas B7 expresses the antidistributive property of conversionrelative to composition.[2]

Converse and composition distribute over disjunction:

B8: (A∨B) ˘ = A ˘ ∨B ˘

B9: (A∨B)•C = (A•C)∨(B•C)

B10 is Tarski’s equational form of the fact, discovered by Augustus De Morgan, that A•B ≤ C− ↔ A ˘ •C ≤ B− ↔C•B ˘ ≤ A−.

B10: (A ˘ •(A•B)−)∨B− = B−

These axioms are ZFC theorems; for the purely BooleanB1-B3, this fact is trivial. After each of the following axiomsis shown the number of the corresponding theorem in chpt. 3 of Suppes (1960), an exposition of ZFC: B4 27, B545, B6 14, B7 26, B8 16, B9 23.

8.2 Expressing properties of binary relations in RA

The following table shows how many of the usual properties of binary relations can be expressed as succinct RAequalities or inequalities. Below, an inequality of the form A≤B is shorthand for the Boolean equation A∨B = B.The most complete set of results of this nature is chpt. C of Carnap (1958), where the notation is rather distant fromthat of this entry. Chpt. 3.2 of Suppes (1960) contains fewer results, presented as ZFC theorems and using a notationthat more resembles that of this entry. Neither Carnap nor Suppes formulated their results using the RA of this entry,or in an equational manner.

8.3 Expressive power

The metamathematics of RA are discussed at length in Tarski and Givant (1987), and more briefly in Givant (2006).RA consists entirely of equations manipulated using nothing more than uniform replacement and the substitution ofequals for equals. Both rules are wholly familiar from school mathematics and from abstract algebra generally. HenceRA proofs are carried out in a manner familiar to all mathematicians, unlike the case in mathematical logic generally.RA can express any (and up to logical equivalence, exactly the) first-order logic (FOL) formulas containing no morethan three variables. (A given variable can be quantified multiple times and hence quantifiers can be nested arbitrarilydeeply by “reusing” variables.) Surprisingly, this fragment of FOL suffices to express Peano arithmetic and almostall axiomatic set theories ever proposed. Hence RA is, in effect, a way of algebraizing nearly all mathematics,

8.4. EXAMPLES 21

while dispensing with FOL and its connectives, quantifiers, turnstiles, and modus ponens. Because RA can expressPeano arithmetic and set theory, Gödel’s incompleteness theorems apply to it; RA is incomplete, incompletable, andundecidable. (N.B. The Boolean algebra fragment of RA is complete and decidable.)The representable relation algebras, forming the class RRA, are those relation algebras isomorphic to some re-lation algebra consisting of binary relations on some set, and closed under the intended interpretation of the RAoperations. It is easily shown, e.g. using the method of pseudoelementary classes, that RRA is a quasivariety, thatis, axiomatizable by a universal Horn theory. In 1950, Roger Lyndon proved the existence of equations holding inRRA that did not hold in RA. Hence the variety generated by RRA is a proper subvariety of the variety RA. In1955, Alfred Tarski showed that RRA is itself a variety. In 1964, Donald Monk showed that RRA has no finiteaxiomatization, unlike RA which is finitely axiomatized by definition.

8.3.1 Q-Relation Algebras

An RA is a Q-Relation Algebra (QRA) if, in addition to B1-B10, there exist some A and B such that (Tarski andGivant 1987: §8.4):

Q0: A ˘ •A ≤ I

Q1: B ˘ •B ≤ I

Q2: A ˘ •B = 1

Essentially these axioms imply that the universe has a (non-surjective) pairing relation whose projections are A andB. It is a theorem that every QRA is a RRA (Proof by Maddux, see Tarski & Givant 1987: 8.4(iii) ).Every QRA is representable (Tarski and Givant 1987). That not every relation algebra is representable is a fun-damental way RA differs from QRA and Boolean algebras which, by Stone’s representation theorem for Booleanalgebras, are always representable as sets of subsets of some set, closed under union, intersection, and complement.

8.4 Examples

1. Any Boolean algebra can be turned into a RA by interpreting conjunction as composition (the monoid multipli-cation •), i.e. x•y is defined as x∧y. This interpretation requires that converse interpret identity (ў = y), and that bothresiduals y\x and x/y interpret the conditional y→x (i.e., ¬y∨x).2. The motivating example of a relation algebra depends on the definition of a binary relation R on a set X as anysubset R ⊆ X², where X² is the Cartesian square of X. The power set 2X² consisting of all binary relations on X isa Boolean algebra. While 2X² can be made a relation algebra by taking R•S = R∧S, as per example (1) above, thestandard interpretation of • is instead x(R•S)z = ∃y.xRySz. That is, the ordered pair (x,z) belongs to the relation R•Sjust when there exists y ∈ X such that (x,y) ∈ R and (y,z) ∈ S. This interpretation uniquely determines R\S as consistingof all pairs (y,z) such that for all x ∈ X, if xRy then xSz. Dually, S/R consists of all pairs (x,y) such that for all z ∈ X, ifyRz then xSz. The translation ў = ¬(y\¬I) then establishes the converse R ˘ of R as consisting of all pairs (y,x) suchthat (x,y) ∈ R.3. An important generalization of the previous example is the power set 2E where E ⊆X² is any equivalence relation onthe set X. This is a generalization because X² is itself an equivalence relation, namely the complete relation consistingof all pairs. While 2E is not a subalgebra of 2X² when E ≠ X² (since in that case it does not contain the relation X²,the top element 1 being E instead of X²), it is nevertheless turned into a relation algebra using the same definitionsof the operations. Its importance resides in the definition of a representable relation algebra as any relation algebraisomorphic to a subalgebra of the relation algebra 2E for some equivalence relation E on some set. The previoussection says more about the relevant metamathematics.4. If group sum or product interprets composition, group inverse interprets converse, group identity interprets I, andif R is a one to one correspondence, so that R ˘ •R = R•R

˘ = I,[3] then L is a group as well as a monoid. B4-B7become well-known theorems of group theory, so that RA becomes a proper extension of group theory as well as ofBoolean algebra.

22 CHAPTER 8. RELATION ALGEBRA

8.5 Historical remarks

DeMorgan founded RA in 1860, but C. S. Peirce took it much further and became fascinated with its philosophicalpower. The work of DeMorgan and Peirce came to be known mainly in the extended and definitive form ErnstSchröder gave it in Vol. 3 of his Vorlesungen (1890–1905). Principia Mathematica drew strongly on Schröder’s RA,but acknowledged him only as the inventor of the notation. In 1912, Alwin Korselt proved that a particular formulain which the quantifiers were nested four deep had no RA equivalent.[4] This fact led to a loss of interest in RA untilTarski (1941) began writing about it. His students have continued to develop RA down to the present day. Tarskireturned to RA in the 1970s with the help of Steven Givant; this collaboration resulted in the monograph by Tarskiand Givant (1987), the definitive reference for this subject. For more on the history ofRA, see Maddux (1991, 2006).

8.6 Software

• RelMICS / Relational Methods in Computer Science maintained by Wolfram Kahl

• Carsten Sinz: ARA / An Automatic Theorem Prover for Relation Algebras

8.7 See also

8.8 Footnotes[1] Alfred Tarski (1948) “Abstract: Representation Problems for Relation Algebras,” Bulletin of the AMS 54: 80.

[2] Chris Brink; Wolfram Kahl; Gunther Schmidt (1997). Relational Methods in Computer Science. Springer. pp. 4 and 8.ISBN 978-3-211-82971-4.

[3] Tarski, A. (1941), p. 87.

[4] Korselt did not publish his finding. It was first published in Leopold Loewenheim (1915) "Über Möglichkeiten im Rela-tivkalkül,” Mathematische Annalen 76: 447–470. Translated as “On possibilities in the calculus of relatives” in Jean vanHeijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 228–251.

8.9 References

• Rudolf Carnap (1958) Introduction to Symbolic Logic and its Applications. Dover Publications.

• Givant, Steven (2006). “The calculus of relations as a foundation for mathematics”. Journal of AutomatedReasoning 37: 277–322. doi:10.1007/s10817-006-9062-x.

• Halmos, P. R., 1960. Naive Set Theory. Van Nostrand.

• Leon Henkin, Alfred Tarski, and Monk, J. D., 1971. Cylindric Algebras, Part 1, and 1985, Part 2. NorthHolland.

• Hirsch R., and Hodkinson, I., 2002, Relation Algebra byGames, vol. 147 in Studies in Logic and the Foundationsof Mathematics. Elsevier Science.

• Jónsson, Bjarni; Tsinakis, Constantine (1993). “Relation algebras as residuated Boolean algebras”. AlgebraUniversalis 30: 469–78. doi:10.1007/BF01195378.

• Maddux, Roger (1991). “The Origin of Relation Algebras in the Development and Axiomatization of theCalculus of Relations” (PDF). Studia Logica 50 (3–4): 421–455. doi:10.1007/BF00370681.

• --------, 2006. Relation Algebras, vol. 150 in Studies in Logic and the Foundations of Mathematics. ElsevierScience.

• Patrick Suppes, 1960. Axiomatic Set Theory. Van Nostrand. Dover reprint, 1972. Chpt. 3.

8.10. EXTERNAL LINKS 23

• Gunther Schmidt, 2010. Relational Mathematics. Cambridge University Press.

• Tarski, Alfred (1941). “On the calculus of relations”. Journal of Symbolic Logic 6: 73–89. doi:10.2307/2268577.

• ------, and Givant, Steven, 1987. A Formalization of Set Theory without Variables. Providence RI: AmericanMathematical Society.

8.10 External links• Yohji AKAMA, Yasuo Kawahara, and Hitoshi Furusawa, "Constructing Allegory from Relation Algebra and

Representation Theorems."

• Richard Bird, Oege de Moor, Paul Hoogendijk, "Generic Programming with Relations and Functors."

• R.P. de Freitas and Viana, "A Completeness Result for Relation Algebra with Binders."

• Peter Jipsen:

• Relation algebras. In Mathematical structures. If there are problems with LaTeX, see an old HTMLversion here.

• "Foundations of Relations and Kleene Algebra."• "Computer Aided Investigations of Relation Algebras."• "A Gentzen System And Decidability For Residuated Lattices.”

• Vaughan Pratt:

• "Origins of the Calculus of Binary Relations." A historical treatment.• "The Second Calculus of Binary Relations."

• Priss, Uta:

• "An FCA interpretation of Relation Algebra."• "Relation Algebra and FCA" Links to publications and software

• Kahl, Wolfram, and Schmidt, Gunther, "Exploring (Finite) Relation Algebras Using Tools Written in Haskell."See homepage of the whole project.

Chapter 9

Relation construction

In logic and mathematics, relation construction and relational constructibility have to do with the ways that onerelation is determined by an indexed family or a sequence of other relations, called the relation dataset. The relationin the focus of consideration is called the faciendum. The relation dataset typically consists of a specified relationover sets of relations, called the constructor, the factor, or the method of construction, plus a specified set of otherrelations, called the faciens, the ingredients, or the makings.Relation composition and relation reduction are special cases of relation constructions.

9.1 See also• Projection

• Relation

• Relation composition

• Relation reduction

24

Chapter 10

Representation (mathematics)

In mathematics, representation is a very general relationship that expresses similarities between objects. Roughlyspeaking, a collection Y of mathematical objects may be said to represent another collection X of objects, providedthat the properties and relationships existing among the representing objects yi conform in some consistent way tothose existing among the corresponding represented objects xi. Somewhat more formally, for a set Π of propertiesand relations, a Π-representation of some structure X is a structure Y that is the image of X under a homomorphismthat preserves Π. The label representation is sometimes also applied to the homomorphism itself.

10.1 Representation theory

Perhaps the most well-developed example of this general notion is the subfield of abstract algebra called representationtheory, which studies the representing of elements of algebraic structures by linear transformations of vector spaces.

10.2 Other examples

Although the term representation theory is well established in the algebraic sense discussed above, there are manyother uses of the term representation throughout mathematics.

10.2.1 Graph theory

An active area of graph theory is the exploration of isomorphisms between graphs and other structures. A keyclass of such problems stems from the fact that, like adjacency in undirected graphs, intersection of sets (or, moreprecisely, non-disjointness) is a symmetric relation. This gives rise to the study of intersection graphs for innumerablefamilies of sets.[1] One foundational result here, due to Paul Erdős and colleagues, is that every n-vertex graph maybe represented in terms of intersection among subsets of a set of size no more than n2/4.[2]

Representing a graph by such algebraic structures as its adjacency matrix and Laplacian matrix gives rise to the fieldof spectral graph theory.[3]

10.2.2 Order theory

Dual to the observation above that every graph is an intersection graph is the fact that every partially ordered set isisomorphic to a collection of sets ordered by the containment (or inclusion) relation ⊆. Among the posets that ariseas the containment orders for natural classes of objects are the Boolean lattices and the orders of dimension n.[4]

Many partial orders arise from (and thus can be represented by) collections of geometric objects. Among them arethe n-ball orders. The 1-ball orders are the interval-containment orders, and the 2-ball orders are the so-called circleorders, the posets representable in terms of containment among disks in the plane. A particularly nice result in thisfield is the characterization of the planar graphs as those graphs whose vertex-edge incidence relations are circleorders.[5]

25

26 CHAPTER 10. REPRESENTATION (MATHEMATICS)

There are also geometric representations that are not based on containment. Indeed, one of the best studied classesamong these are the interval orders,[6] which represent the partial order in terms of what might be called disjointprecedence of intervals on the real line: each element x of the poset is represented by an interval [x1, x2] such that forany y and z in the poset, y is below z if and only if y2 < z1.

10.2.3 Polysemy

Under certain circumstances, a single function f:X → Y is at once an isomorphism from several mathematical struc-tures on X. Since each of those structures may be thought of, intuitively, as a meaning of the image Y—one of thethings that Y is trying to tell us—this phenomenon is called polysemy, a term borrowed from linguistics. Examplesinclude:

• intersection polysemy—pairs of graphs G1 and G2 on a common vertex set V that can be simultaneouslyrepresented by a single collection of sets Sv such that any distinct vertices u and w in V...

are adjacent in G1 if and only if their corresponding sets intersect ( Su ∩ Sw ≠ Ø ), andare adjacent in G2 if and only if the complements do ( SuC ∩ SwC ≠ Ø ).[7]

• competition polysemy—motivated by the study of ecological food webs, in which pairs of species may haveprey in common or have predators in common. A pair of graphs G1 and G2 on one vertex set is competitionpolysemic if and only if there exists a single directed graph D on the same vertex set such that any distinctvertices u and v...

are adjacent in G1 if and only if there is a vertex w such that both uw and vw are arcs in D,andare adjacent in G2 if and only if there is a vertex w such that both wu and wv are arcs in D.[8]

• interval polysemy—pairs of posets P1 and P2 on a common ground set that can be simultaneously representedby a single collection of real intervals that is an interval-order representation of P1 and an interval-containmentrepresentation of P2.[9]

10.3 See also• Representation theorems

• Model theory

10.4 References[1] • McKee, Terry A.; McMorris, F. R. (1999), Topics in Intersection Graph Theory, SIAM Monographs on Discrete

Mathematics and Applications, Philadelphia: Society for Industrial and Applied Mathematics, ISBN 0-89871-430-3, MR 1672910

[2] Erdős, Paul; Goodman, A. W.; Pósa, Louis (1966), “The representation of a graph by set intersections”, Canadian Journalof Mathematics 18 (1): 106–112, doi:10.4153/cjm-1966-014-3, MR 0186575

[3] • Biggs, Norman (1994), Algebraic Graph Theory, Cambridge Mathematical Library, Cambridge University Press,ISBN 978-0-521-45897-9, MR 1271140

[4] • Trotter, William T. (1992), Combinatorics and Partially Ordered Sets: Dimension Theory, Johns Hopkins Series in theMathematical Sciences, Baltimore: The Johns Hopkins University Press, ISBN 978-0-8018-4425-6, MR 1169299

[5] • Scheinerman, Edward (1991), “A note on planar graphs and circle orders”, SIAM Journal on Discrete Mathematics 4(3): 448–451, doi:10.1137/0404040, MR 1105950

[6] • Fishburn, Peter C. (1985), Interval Orders and Interval Graphs: A Study of Partially Ordered Sets, Wiley-InterscienceSeries in Discrete Mathematics, John Wiley & Sons, ISBN 978-0-471-81284-5, MR 0776781

10.4. REFERENCES 27

[7] • Tanenbaum, Paul J. (1999), “Simultaneous intersection representation of pairs of graphs”, Journal of Graph Theory32 (2): 171–190, doi:10.1002/(SICI)1097-0118(199910)32:2<171::AID-JGT7>3.0.CO;2-N, MR 1709659

[8] • Fischermann, Miranca; Knoben, Werner; Kremer, Dirk; Rautenbachh, Dieter (2004), “Competition polysemy”,Discrete Mathematics 282 (1–3): 251–255, doi:10.1016/j.disc.2003.11.014, MR 2059526

[9] • Tanenbaum, Paul J. (1996), “Simultaneous representation of interval and interval-containment orders”, Order 13 (4):339–350, doi:10.1007/BF00405593, MR 1452517

Chapter 11

Separoid

In mathematics, a separoid is a binary relation between disjoint sets which is stable as an ideal in the canonical orderinduced by inclusion. Many mathematical objects which appear to be quite different, find a common generalisation inthe framework of separoids; e.g., graphs, configurations of convex sets, oriented matroids, and polytopes. Any count-able category is an induced subcategory of separoids when they are endowed with homomorphisms (viz., mappingsthat preserve the so-called minimal Radon partitions).In this general framework, some results and invariants of different categories turn out to be special cases of thesame aspect; e.g., the pseudoachromatic number from graph theory and the Tverberg theorem from combinatorialconvexity are simply two faces of the same aspect, namely, complete colouring of separoids.

11.1 The axioms

A separoid is a set S endowed with a binary relation | ⊆ 2S × 2S on its power set, which satisfies the followingsimple properties for A,B ⊆ S :

A | B ⇔ B | A,

A | B ⇒ A ∩B = ∅,

A | B and A′ ⊂ A ⇒ A′ | B.

A related pair A | B is called a separation and we often say that A is separated from B. It is enough to know themaximal separations to reconstruct the separoid.A mapping φ : S → T is a morphism of separoids if the preimages of separations are separations; that is, forA,B ⊆ T

A | B ⇒ φ−1(A) | φ−1(B).

11.2 Examples

Examples of separoids can be found in almost every branch of mathematics. Here we list just a few.1. Given a graph G=(V,E), we can define a separoid on its vertices by saying that two (disjoint) subsets of V, say Aand B, are separated if there are no edges going from one to the other; i.e.,

A | B ⇔ ∀a ∈ A and b ∈ B : ab ̸∈ E.

28

11.3. THE BASIC LEMMA 29

2. Given an oriented matroid M = (E,T), given in terms of its topes T, we can define a separoid on E by saying thattwo subsets are separated if they are contained in opposite signs of a tope. In other words, the topes of an orientedmatroid are the maximal separations of a separoid. This example includes, of course, all directed graphs.3. Given a family of objects in an Euclidean space, we can define a separoid in it by saying that two subsets areseparated if there exists a hyperplane that separates them; i.e., leaving them in the two opposite sides of it.4. Given a topological space, we can define a separoid saying that two subsets are separated if there exist two disjointopen sets which contains them (one for each of them).

11.3 The basic lemma

Every separoid can be represented with a family of convex sets in some Euclidean space and their separations byhyperplanes.

11.4 References• Strausz Ricardo; “Separoides”. Situs, serie B, no. 5 (1998), Universidad Nacional Autónoma de México.

• Arocha Jorge Luis, Bracho Javier, Montejano Luis, Oliveros Deborah, Strausz Ricardo; “Separoids, their cat-egories and a Hadwiger-type theorem for transversals”. Discrete and Computational Geometry 27 (2002), no.3, 377–385.

• Strausz Ricardo; “Separoids and a Tverberg-type problem”. Geombinatorics 15 (2005), no. 2, 79–92.

• Montellano-Ballesteros Juan Jose, Por Attila, Strausz Ricardo; “Tverberg-type theorems for separoids”. Dis-crete and Computational Geometry 35 (2006), no.3, 513–523.

• Nešetřil Jaroslav, Strausz Ricardo; “Universality of separoids”. ArchivumMathematicum (Brno) 42 (2006), no.1, 85–101.

• Bracho Javier, Strausz Ricardo; “Two geometric representations of separoids”. Periodica Mathematica Hun-garica 53 (2006), no. 1-2, 115–120.

• Strausz Ricardo; “Homomorphisms of separoids”. 6th Czech-Slovak International Symposium on Combina-torics, Graph Theory, Algorithms and Applications, 461–468, Electronic Notes on Discrete Mathematics 28,Elsevier, Amsterdam, 2007.

• Strausz Ricardo; “Edrös-Szekeres 'happy end'-type theorems for separoids”. European Journal of Combina-torics 29 (2008), no. 4, 1076–1085.

Chapter 12

Series-parallel partial order

Series composition

Parallel composition

A series-parallel partial order, shown as a Hasse diagram.

In order-theoretic mathematics, a series-parallel partial order is a partially ordered set built up from smaller series-

30

12.1. DEFINITION 31

parallel partial orders by two simple composition operations.[1][2]

The series-parallel partial orders may be characterized as the N-free finite partial orders; they have order dimension atmost two.[1][3] They include weak orders and the reachability relationship in directed trees and directed series-parallelgraphs.[2][3] The comparability graphs of series-parallel partial orders are cographs.[2][4]

Series-parallel partial orders have been applied in job shop scheduling,[5] machine learning of event sequencing intime series data,[6] transmission sequencing of multimedia data,[7] and throughput maximization in dataflow pro-gramming.[8]

Series-parallel partial orders have also been called multitrees;[4] however, that name is ambiguous: multitrees alsorefer to partial orders with no four-element diamond suborder[9] and to other structures formed from multiple trees.

12.1 Definition

Consider P and Q, two partially ordered sets. The series composition of P and Q, written P; Q,[7] P * Q,[2] or P ⧀Q,[1]is the partially ordered set whose elements are the disjoint union of the elements of P andQ. In P;Q, two elementsx and y that both belong to P or that both belong to Q have the same order relation that they do in P or Q respectively.However, for every pair x, y where x belongs to P and y belongs to Q, there is an additional order relation x ≤ y in theseries composition. Series composition is an associative operation: one can write P; Q; R as the series compositionof three orders, without ambiguity about how to combine them pairwise, because both of the parenthesizations (P;Q); R and P; (Q; R) describe the same partial order. However, it is not a commutative operation, because switchingthe roles of P and Q will produce a different partial order that reverses the order relations of pairs with one elementin P and one in Q.[1]

The parallel composition of P and Q, written P || Q,[7] P + Q,[2] or P ⊕ Q,[1] is defined similarly, from the disjointunion of the elements in P and the elements in Q, with pairs of elements that both belong to P or both to Q havingthe same order as they do in P or Q respectively. In P || Q, a pair x, y is incomparable whenever x belongs to P and ybelongs to Q. Parallel composition is both commutative and associative.[1]

The class of series-parallel partial orders is the set of partial orders that can be built up from single-element partialorders using these two operations. Equivalently, it is the smallest set of partial orders that includes the single-elementpartial order and is closed under the series and parallel composition operations.[1][2]

A weak order is the series parallel partial order obtained from a sequence of composition operations in which allof the parallel compositions are performed first, and then the results of these compositions are combined using onlyseries compositions.[2]

12.2 Forbidden suborder characterization

The partial order N with the four elements a, b, c, and d and exactly the three order relations a ≤ b ≥ c ≤ d is anexample of a fence or zigzag poset; its Hasse diagram has the shape of the capital letter “N”. It is not series-parallel,because there is no way of splitting it into the series or parallel composition of two smaller partial orders. A partialorder P is said to be N-free if there does not exist a set of four elements in P such that the restriction of P to thoseelements is order-isomorphic to N. The series-parallel partial orders are exactly the nonempty finite N-free partialorders.[1][2][3]

It follows immediately from this (although it can also be proven directly) that any nonempty restriction of a series-parallel partial order is itself a series-parallel partial order.[1]

12.3 Order dimension

The order dimension of a partial order P is the minimum size of a realizer of P, a set of linear extensions of P withthe property that, for every two distinct elements x and y of P, x ≤ y in P if and only if x has an earlier position thany in every linear extension of the realizer. Series-parallel partial orders have order dimension at most two. If P andQ have realizers {L1, L2} and {L3, L4}, respectively, then {L1L3, L2L4} is a realizer of the series composition P; Q,and {L1L3, L4L2} is a realizer of the parallel composition P || Q.[2][3] A partial order is series-parallel if and only ifit has a realizer in which one of the two permutations is the identity and the other is a separable permutation.

32 CHAPTER 12. SERIES-PARALLEL PARTIAL ORDER

It is known that a partial order P has order dimension two if and only if there exists a conjugate order Q on thesame elements, with the property that any two distinct elements x and y are comparable on exactly one of these twoorders. In the case of series parallel partial orders, a conjugate order that is itself series parallel may be obtained byperforming a sequence of composition operations in the same order as the ones defining P on the same elements,but performing a series composition for each parallel composition in the decomposition of P and vice versa. Morestrongly, although a partial order may have many different conjugates, every conjugate of a series parallel partial ordermust itself be series parallel.[2]

12.4 Connections to graph theory

Any partial order may be represented (usually in more than one way) by a directed acyclic graph in which there is apath from x to ywhenever x and y are elements of the partial order with x ≤ y. The graphs that represent series-parallelpartial orders in this way have been called vertex series parallel graphs, and their transitive reductions (the graphsof the covering relations of the partial order) are called minimal vertex series parallel graphs.[3] Directed trees and(two-terminal) series parallel graphs are examples of minimal vertex series parallel graphs; therefore, series parallelpartial orders may be used to represent reachability relations in directed trees and series parallel graphs.[2][3]

The comparability graph of a partial order is the undirected graph with a vertex for each element and an undirectededge for each pair of distinct elements x, y with either x ≤ y or y ≤ x. That is, it is formed from a minimal vertexseries parallel graph by forgetting the orientation of each edge. The comparability graph of a series-partial order is acograph: the series and parallel composition operations of the partial order give rise to operations on the comparabilitygraph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these twooperations are the basic operations from which cographs are defined. Conversely, every cograph is the comparabilitygraph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be aseries-parallel partial order, because every other kind of partial order has an N suborder that would correspond to aninduced four-vertex path in its comparability graph, and such paths are forbidden in cographs.[2][4]

12.5 Computational complexity

It is possible to use the forbidden suborder characterization of series-parallel partial orders as a basis for an algorithmthat tests whether a given binary relation is a series-parallel partial order, in an amount of time that is linear inthe number of related pairs.[2][3] Alternatively, if a partial order is described as the reachability order of a directedacyclic graph, it is possible to test whether it is a series-parallel partial order, and if so compute its transitive closure,in time proportional to the number of vertices and edges in the transitive closure; it remains open whether the timeto recognize series-parallel reachability orders can be improved to be linear in the size of the input graph.[10]

If a series-parallel partial order is represented as an expression tree describing the series and parallel compositionoperations that formed it, then the elements of the partial order may be represented by the leaves of the expressiontree. A comparison between any two elements may be performed algorithmically by searching for the lowest commonancestor of the corresponding two leaves; if that ancestor is a parallel composition, the two elements are incomparable,and otherwise the order of the series composition operands determines the order of the elements. In this way, a series-parallel partial order on n elements may be represented in O(n) space with O(1) time to determine any comparisonvalue.[2]

It is NP-complete to test, for two given series-parallel partial orders P and Q, whether P contains a restriction iso-morphic to Q.[3]

Although the problem of counting the number of linear extensions of an arbitrary partial order is #P-complete,[11] itmay be solved in polynomial time for series-parallel partial orders. Specifically, if L(P) denotes the number of linearextensions of a partial order P, then L(P; Q) = L(P)L(Q) and

L(P ||Q) =(|P |+ |Q|)!|P |!|Q|!

L(P )L(Q),

so the number of linear extensions may be calculated using an expression tree with the same form as the decompositiontree of the given series-parallel order.[2]

12.6. APPLICATIONS 33

12.6 Applications

Mannila & Meek (2000) use series-parallel partial orders as a model for the sequences of events in time series data.They describe machine learning algorithms for inferring models of this type, and demonstrate its effectiveness atinferring course prerequisites from student enrollment data and at modeling web browser usage patterns.[6]

Amer et al. (1994) argue that series-parallel partial orders are a good fit for modeling the transmission sequencingrequirements of multimedia presentations. They use the formula for computing the number of linear extensions of aseries-parallel partial order as the basis for analyzing multimedia transmission algorithms.[7]

Choudhary et al. (1994) use series-parallel partial orders to model the task dependencies in a dataflow model ofmassive data processing for computer vision. They show that, by using series-parallel orders for this problem, it ispossible to efficiently construct an optimized schedule that assigns different tasks to different processors of a parallelcomputing system in order to optimize the throughput of the system.[8]

A class of orderings somewhat more general than series-parallel partial orders is provided by PQ trees, data structuresthat have been applied in algorithms for testing whether a graph is planar and recognizing interval graphs.[12] A Pnode of a PQ tree allows all possible orderings of its children, like a parallel composition of partial orders, while a Qnode requires the children to occur in a fixed linear ordering, like a series composition of partial orders. However,unlike series-parallel partial orders, PQ trees allow the linear ordering of any Q node to be reversed.

12.7 See also• Series and parallel circuits

12.8 References[1] Bechet, Denis; De Groote, Philippe; Retoré, Christian (1997), “A complete axiomatisation for the inclusion of series-

parallel partial orders”, Rewriting Techniques and Applications, Lecture Notes in Computer Science 1232, Springer-Verlag,pp. 230–240, doi:10.1007/3-540-62950-5_74.

[2] Möhring, Rolf H. (1989), “Computationally tractable classes of ordered sets”, in Rival, Ivan, Algorithms and Order: Pro-ceedings of the NATO Advanced Study Institute on Algorithms and Order, Ottawa, Canada, May 31-June 13, 1987, NATOScience Series C 255, Springer-Verlag, pp. 105–194, ISBN 978-0-7923-0007-6.

[3] Valdes, Jacobo; Tarjan, Robert E.; Lawler, Eugene L. (1982), “The recognition of series parallel digraphs”, SIAM Journalon Computing 11 (2): 298–313, doi:10.1137/0211023.

[4] Jung, H. A. (1978), “On a class of posets and the corresponding comparability graphs”, Journal of Combinatorial Theory,Series B 24 (2): 125–133, doi:10.1016/0095-8956(78)90013-8, MR 0491356.

[5] Lawler, Eugene L. (1978), “Sequencing jobs to minimize total weighted completion time subject to precedence constraints”,Annals of Discrete Mathematics 2: 75–90, doi:10.1016/S0167-5060(08)70323-6, MR 0495156.

[6] Mannila, Heikki; Meek, Christopher (2000), “Global partial orders from sequential data”, Proc. 6th ACM SIGKDD Inter-national Conference on Knowledge Discovery and Data Mining (KDD 2000), pp. 161–168, doi:10.1145/347090.347122.

[7] Amer, Paul D.; Chassot, Christophe; Connolly, Thomas J.; Diaz, Michel; Conrad, Phillip (1994), “Partial-order transportservice for multimedia and other applications”, IEEE/ACMTransactions onNetworking 2 (5): 440–456, doi:10.1109/90.336326.

[8] Choudhary, A. N.; Narahari, B.; Nicol, D. M.; Simha, R. (1994), “Optimal processor assignment for a class of pipelinedcomputations”, IEEE Transactions on Parallel and Distributed Systems 5 (4): 439–445, doi:10.1109/71.273050.

[9] Furnas, George W.; Zacks, Jeff (1994), “Multitrees: enriching and reusing hierarchical structure”, Proc. SIGCHI conferenceon Human Factors in Computing Systems (CHI '94), pp. 330–336, doi:10.1145/191666.191778.

[10] Ma, Tze-Heng; Spinrad, Jeremy (1991), “Transitive closure for restricted classes of partial orders”, Order 8 (2): 175–183,doi:10.1007/BF00383402.

[11] Brightwell, Graham R.; Winkler, Peter (1991), “Counting linear extensions”,Order 8 (3): 225–242, doi:10.1007/BF00383444.

[12] Booth, Kellogg S.; Lueker, George S. (1976), “Testing for the consecutive ones property, interval graphs, and graphplanarity using PQ-tree algorithms”, Journal of Computer and System Sciences 13 (3): 335–379, doi:10.1016/S0022-0000(76)80045-1.

Chapter 13

Surjective function

“Onto” redirects here. For other uses, see wikt:onto.In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if every element y in Y

X1

2

3

4

YD

B

C

A surjective function from domain X to codomain Y. The function is surjective because every point in the codomain is the value off(x) for at least one point x in the domain.

34

13.1. DEFINITION 35

has a corresponding element x in X such that f(x) = y. The function f may map more than one element of X to thesame element of Y.The term surjective and the related terms injective and bijectivewere introduced by Nicolas Bourbaki,[1] the pseudonymfor a group of mainly French 20th-century mathematicians who wrote a series of books presenting an exposition ofmodern advanced mathematics, beginning in 1935. The French prefix sur means over or above and relates to the factthat the image of the domain of a surjective function completely covers the function’s codomain.

13.1 Definition

For more details on notation, see Function (mathematics) § Notation.

A surjective function is a function whose image is equal to its codomain. Equivalently, a function f with domainX and codomain Y is surjective if for every y in Y there exists at least one x in X with f(x) = y . Surjections aresometimes denoted by a two-headed rightwards arrow (U+21A0 ↠ rightwards two headed arrow),[2] as in f : X ↠Y.Symbolically,

If f : X → Y , then f is said to be surjective if

∀y ∈ Y, ∃x ∈ X, f(x) = y

13.2 Examples

For any set X, the identity function idX on X is surjective.The function f : Z → {0,1} defined by f(n) = n mod 2 (that is, even integers are mapped to 0 and odd integers to 1)is surjective.The function f : R → R defined by f(x) = 2x + 1 is surjective (and even bijective), because for every real number ywe have an x such that f(x) = y: an appropriate x is (y − 1)/2.The function f : R → R defined by f(x) = x3 − 3x is surjective, because the pre-image of any real number y is thesolution set of the cubic polynomial equation x3 − 3x − y = 0 and every cubic polynomial with real coefficients has atleast one real root. However, this function is not injective (and hence not bijective) since e.g. the pre-image of y = 2is {x = −1, x = 2}. (In fact, the pre-image of this function for every y, −2 ≤ y ≤ 2 has more than one element.)The function g : R → R defined by g(x) = x2 is not surjective, because there is no real number x such that x2 = −1.However, the function g : R → R0

+ defined by g(x) = x2 (with restricted codomain) is surjective because for every yin the nonnegative real codomain Y there is at least one x in the real domain X such that x2 = y.The natural logarithm function ln : (0,+∞) → R is a surjective and even bijective mapping from the set of positivereal numbers to the set of all real numbers. Its inverse, the exponential function, is not surjective as its range is the setof positive real numbers and its domain is usually defined to be the set of all real numbers. The matrix exponentialis not surjective when seen as a map from the space of all n×n matrices to itself. It is, however, usually defined as amap from the space of all n×n matrices to the general linear group of degree n, i.e. the group of all n×n invertiblematrices. Under this definition the matrix exponential is surjective for complex matrices, although still not surjectivefor real matrices.The projection from a cartesian product A × B to one of its factors is surjective unless the other factor is empty.In a 3D video game vectors are projected onto a 2D flat screen by means of a surjective function.

13.3 Properties

A function is bijective if and only if it is both surjective and injective.

36 CHAPTER 13. SURJECTIVE FUNCTION

X Y

f(x)

f : X → Y

x

A non-surjective function from domain X to codomain Y. The smaller oval inside Y is the image (also called range) of f. Thisfunction is not surjective, because the image does not fill the whole codomain. In other words, Y is colored in a two-step process:First, for every x in X, the point f(x) is colored yellow; Second, all the rest of the points in Y, that are not yellow, are colored blue.The function f is surjective only if there are no blue points.

If (as is often done) a function is identified with its graph, then surjectivity is not a property of the function itself, butrather a relationship between the function and its codomain. Unlike injectivity, surjectivity cannot be read off of thegraph of the function alone.

13.3.1 Surjections as right invertible functions

The function g : Y → X is said to be a right inverse of the function f : X → Y if f(g(y)) = y for every y in Y (g can beundone by f). In other words, g is a right inverse of f if the composition f o g of g and f in that order is the identityfunction on the domain Y of g. The function g need not be a complete inverse of f because the composition in theother order, g o f, may not be the identity function on the domain X of f. In other words, f can undo or "reverse" g,but cannot necessarily be reversed by it.Every function with a right inverse is necessarily a surjection. The proposition that every surjective function has aright inverse is equivalent to the axiom of choice.If f : X → Y is surjective and B is a subset of Y, then f(f −1(B)) = B. Thus, B can be recovered from its preimagef −1(B).For example, in the first illustration, there is some function g such that g(C) = 4. There is also some function f suchthat f(4) = C. It doesn't matter that g(C) can also equal 3; it only matters that f “reverses” g.

• Another surjective function. (This one happens to be a bijection)

• A non-surjective function. (This one happens to be an injection)

• Surjective composition: the first function need not be surjective.

13.3. PROPERTIES 37

x

y

x

y

1 X 1 Y : f 2 X 2 Y : f

2 X x 1 X x

x f y

f im Y y

Y y

f im

Y X : f x f y

X x

Interpretation for surjective functions in the Cartesian plane, defined by the mapping f : X → Y, where y = f(x), X = domain offunction, Y = range of function. Every element in the range is mapped onto from an element in the domain, by the rule f. Theremay be a number of domain elements which map to the same range element. That is, every y in Y is mapped from an element x inX, more than one x can map to the same y. Left: Only one domain is shown which makes f surjective. Right: two possible domainsX1 and X2 are shown.

x

y

X x 0

Y X : f x f y

X x

Y y

f im

Y y 0

x

y

X x 1

Y y 2

Y y 1

X x 2

X x 3

Y y 3

X x

1 X 1 Y : f 2 X 2 Y : f

Y y

f im

2 X x 1 X x

x f y

Non-surjective functions in the Cartesian plane. Although some parts of the function are surjective, where elements y in Y do havea value x in X such that y = f(x), some parts are not. Left: There is y0 in Y, but there is no x0 in X such that y0 = f(x0). Right:There are y1, y2 and y3 in Y, but there are no x1, x2, and x3 in X such that y1 = f(x1), y2 = f(x2), and y3 = f(x3).

13.3.2 Surjections as epimorphisms

A function f : X → Y is surjective if and only if it is right-cancellative:[3] given any functions g,h : Y → Z, whenever go f = h o f, then g = h. This property is formulated in terms of functions and their composition and can be generalizedto the more general notion of the morphisms of a category and their composition. Right-cancellative morphisms arecalled epimorphisms. Specifically, surjective functions are precisely the epimorphisms in the category of sets. The

38 CHAPTER 13. SURJECTIVE FUNCTION

prefix epi is derived from the Greek preposition ἐπί meaning over, above, on.Any morphism with a right inverse is an epimorphism, but the converse is not true in general. A right inverse g of amorphism f is called a section of f. A morphism with a right inverse is called a split epimorphism.

13.3.3 Surjections as binary relations

Any function with domain X and codomain Y can be seen as a left-total and right-unique binary relation between Xand Y by identifying it with its function graph. A surjective function with domain X and codomain Y is then a binaryrelation between X and Y that is right-unique and both left-total and right-total.

13.3.4 Cardinality of the domain of a surjection

The cardinality of the domain of a surjective function is greater than or equal to the cardinality of its codomain: If f: X → Y is a surjective function, then X has at least as many elements as Y, in the sense of cardinal numbers. (Theproof appeals to the axiom of choice to show that a function g : Y → X satisfying f(g(y)) = y for all y in Y exists. gis easily seen to be injective, thus the formal definition of |Y | ≤ |X| is satisfied.)Specifically, if both X and Y are finite with the same number of elements, then f : X → Y is surjective if and only iff is injective.

13.3.5 Composition and decomposition

The composite of surjective functions is always surjective: If f and g are both surjective, and the codomain of gis equal to the domain of f, then f o g is surjective. Conversely, if f o g is surjective, then f is surjective (but g,the function applied first, need not be). These properties generalize from surjections in the category of sets to anyepimorphisms in any category.Any function can be decomposed into a surjection and an injection: For any function h : X→ Z there exist a surjectionf : X → Y and an injection g : Y → Z such that h = g o f. To see this, define Y to be the sets h −1(z) where z is inZ. These sets are disjoint and partition X. Then f carries each x to the element of Y which contains it, and g carrieseach element of Y to the point in Z to which h sends its points. Then f is surjective since it is a projection map, andg is injective by definition.

13.3.6 Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijectiondefined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. More precisely, everysurjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalenceclasses of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set ofall preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class[x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~).

13.4 See also• Bijection, injection and surjection

• Cover (algebra)

• Covering map

• Enumeration

• Fiber bundle

• Index set

• Section (category theory)

13.5. NOTES 39

13.5 Notes[1] “Injection, Surjection and Bijection”, Earliest Uses of Some of the Words of Mathematics, Tripod.

[2] “Arrows – Unicode” (PDF). Retrieved 2013-05-11.

[3] Goldblatt, Robert (2006) [1984]. Topoi, the Categorial Analysis of Logic (Revised ed.). Dover Publications. ISBN 978-0-486-45026-1. Retrieved 2009-11-25.

13.6 References• Bourbaki, Nicolas (2004) [1968]. Theory of Sets. Springer. ISBN 978-3-540-22525-6.

Chapter 14

Symmetric closure

In mathematics, the symmetric closure of a binary relation R on a set X is the smallest symmetric relation on X thatcontains R.For example, if X is a set of airports and xRy means “there is a direct flight from airport x to airport y", then thesymmetric closure of R is the relation “there is a direct flight either from x to y or from y to x". Or, if X is the set ofhumans (alive or dead) and R is the relation 'parent of', then the symmetric closure of R is the relation "x is a parentor a child of y".

14.1 Definition

The symmetric closure S of a relation R on a set X is given by

S = R ∪ {(x, y) : (y, x) ∈ R} .

In other words, the symmetric closure of R is the union of R with its inverse relation, R−1.

14.2 See also• Transitive closure

• Reflexive closure

14.3 References• Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8

40

Chapter 15

Symmetric relation

In mathematics and other areas, a binary relation R over a set X is symmetric if it holds for all a and b in X that if ais related to b then b is related to a.In mathematical notation, this is:

∀a, b ∈ X, aRb ⇒ bRa.

15.1 Examples

15.1.1 In mathematics

• “is equal to” (equality) (whereas “is less than” is not symmetric)

• “is comparable to”, for elements of a partially ordered set

• "... and ... are odd":

41

42 CHAPTER 15. SYMMETRIC RELATION

15.1.2 Outside mathematics

• “is married to” (in most legal systems)

• “is a fully biological sibling of”

• “is a homophone of”

15.2 Relationship to asymmetric and antisymmetric relations

By definition, a relation cannot be both symmetric and asymmetric (where if a is related to b, then b cannot be relatedto a (in the same way)). However, a relation can be neither symmetric nor asymmetric, which is the case for “is lessthan or equal to” and “preys on”).Symmetric and antisymmetric (where the only way a can be related to b and b be related to a is if a = b) are actuallyindependent of each other, as these examples show.

15.3 Additional aspects

A symmetric relation that is also transitive and reflexive is an equivalence relation.One way to conceptualize a symmetric relation in graph theory is that a symmetric relation is an edge, with the edge’stwo vertices being the two entities so related. Thus, symmetric relations and undirected graphs are combinatoriallyequivalent objects.

15.4. SEE ALSO 43

15.4 See also• Symmetry in mathematics

• Symmetry

• Asymmetric relation

• Antisymmetric relation

44 CHAPTER 15. SYMMETRIC RELATION

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