Algebra Sc

pdf version of the entry

Algebrahttp://plato.stanford.edu/archives/spr2014/entries/algebra/

from the Spring 2014 Edition of the

Stanford Encyclopedia

of Philosophy

Edward N. Zalta Uri Nodelman Colin Allen R. Lanier Anderson

Principal Editor Senior Editor Associate Editor Faculty Sponsor

Editorial Board

http://plato.stanford.edu/board.html

Library of Congress Catalog Data

ISSN: 1095-5054

Notice: This PDF version was distributed by request to mem-

bers of the Friends of the SEP Society and by courtesy to SEP

content contributors. It is solely for their fair use. Unauthorized

distribution is prohibited. To learn how to join the Friends of the

SEP Society and obtain authorized PDF versions of SEP entries,

please visit https://leibniz.stanford.edu/friends/ .

Stanford Encyclopedia of Philosophy

Copyright c© 2013 by the publisher

The Metaphysics Research Lab

Center for the Study of Language and Information

Stanford University, Stanford, CA 94305

Algebra

Copyright c© 2014 by the author

Vaughan Pratt

All rights reserved.

Copyright policy: https://leibniz.stanford.edu/friends/info/copyright/

AlgebraFirst published Tue May 29, 2007

Algebra is a branch of mathematics sibling to geometry, analysis(calculus), number theory, combinatorics, etc. Although algebra has itsroots in numerical domains such as the reals and the complex numbers, inits full generality it differs from its siblings in serving no specificmathematical domain. Whereas geometry treats spatial entities, analysiscontinuous variation, number theory integer arithmetic, and combinatoricsdiscrete structures, algebra is equally applicable to all these and othermathematical domains.

Elementary algebra, in use for centuries and taught in secondary school, isthe arithmetic of indefinite quantities or variables x, y, …. Whereas thedefinite sum 3+4 evaluates to the definite quantity 7, the indefinite sumx+y has no definite value, yet we can still say that it is always equal toy+x, or to x²−y² if and only if x is either −y or y+1.

Elementary algebra provides finite ways of managing the infinite. Aformula such as πr² for the area of a circle of radius r describes infinitelymany possible computations, one for each possible valuation of itsvariables. A universally true law expresses infinitely many cases, forexample the single equation x+y = y+x summarizes the infinitely manyfacts 1+2 = 2+1, 3+7 = 7+3, etc. The equation 2x = 4 selects one numberfrom an infinite set of possibilities. And y = 2x+3 expresses the infinitelymany points of the line with slope 2 passing through (0, 3) with a finiteequation whose solutions are exactly those points.

Elementary algebra ordinarily works with real or complex values.However its general methods, if not always its specific operations andlaws, are equally applicable to other numeric domains such as the natural

1

numbers, the integers, the integers modulo some integer n, the rationals,the quaternions, the Gaussian integers, the p-adic numbers, and so on.They are also applicable to many nonnumeric domains such as the subsetsof a given set under the operations of union and intersection, the wordsover a given alphabet under the operations of concatenation and reversal,the permutations of a given set under the operations of composition andinverse, etc. Each such algebraic structure, or simply algebra, consists ofthe set of its elements and operations on those elements obeying the lawsholding in that domain, such as the set Z = {0, ±1, ±2, …} of integersunder the integer operations x+y of addition, xy of multiplication, and −x,negation, or the set 2X of subsets of a set X under the set operations X∪Yof union, X∩Y of intersection, and Xʹ′, complement relative to X.

The laws are often similar but not identical. For example integermultiplication distributes over addition, x(y+z) = xy+xz, but notconversely, for example 2+(3 × 5) = 17 but (2+3) × (2+5) = 35. In theanalogy that makes intersection the set theoretic counterpart ofmultiplication and union that of addition, intersection distributes overunion, X∩(Y∪Z) = (X∩Y)∪(X∩Z), as for the integers, but unlike theintegers union also distributes over intersection: X∪(Y∩Z) =(X∪Y)∩(X∪Z).

Whereas elementary algebra is conducted in a fixed algebra, abstract ormodern algebra treats classes of algebras having certain properties incommon, typically those expressible as equations. The subject, whichemerged during the 19th century, is traditionally introduced via the classesof groups, rings, and fields. For example any number system under theoperations of addition and subtraction forms an abelian (commutative)group; one then passes to rings by bringing in multiplication, and furtherto fields with division. The common four-function calculator provides thefour functions of the field of reals.

Algebra

2 Stanford Encyclopedia of Philosophy

The abstract concept of group in full generality is defined not in terms of aset of numbers but rather as an arbitrary set equipped with a binaryoperation xy, a unary inverse x−1 of that operation, and a unit e satisfyingcertain equations characteristic of groups. One striking novelty withgroups not encountered in everyday elementary algebra is that theirmultiplication need not be abelian: xy and yx can be different! For examplethe group S3 of the six possible permutations of three things is not abelian,as can be seen by exchanging adjacent pairs of letters in the word dan. Ifyou exchange the two letters on the left before the two on the right you getadn and then and, but if you perform these exchanges in the other orderyou get dna and then nda instead of and. Likewise the group of43,252,003,274,489,856,000 operations on Rubik's cube and the infinitegroup SO(3) of rotations of the sphere are not abelian, though the infinitegroup SO(2) of rotations of the circle is abelian. Quaternion multiplicationand matrix multiplication is also noncommutative. Abelian groups areoften called additive groups and their group operation is referred to asaddition x+y rather than multiplication xy.

Groups, rings and fields only scratch the surface of abstract algebra.Vector spaces and more generally modules are restricted forms of rings inwhich the operands of multiplication are required to be a scalar and avector. Monoids generalize groups by dropping inverse; for example thenatural numbers form a monoid but not a group for want of negation.Boolean algebras abstract the algebra of sets. Lattices generalize Booleanalgebras by dropping complement and the distributivity laws.

A number of branches of mathematics have found algebra such aneffective tool that they have spawned algebraic subbranches. Algebraiclogic, algebraic number theory, and algebraic topology are all heavilystudied, while algebraic geometry and algebraic combinatorics have entirejournals devoted to them.

Vaughan Pratt

Spring 2014 Edition 3

Algebra is of philosophical interest for at least two reasons. From theperspective of foundations of mathematics, algebra is strikingly differentfrom other branches of mathematics in both its domain independence andits close affinity to formal logic. Furthermore the dichotomy betweenelementary and abstract algebra reflects a certain duality in reasoning thatDescartes, the inventor of Cartesian Dualism, would have appreciated,wherein the former deals with the reasoning process and the latter thatwhich is reasoned about, as respectively the mind and body ofmathematics.

Algebra has also played a significant role in clarifying and highlightingnotions of logic, at the core of exact philosophy for millennia. The firststep away from the Aristotelian logic of syllogisms towards a morealgebraic form of logic was taken by Boole in an 1847 pamphlet andsubsequently in a more detailed treatise, The Laws of Thought, in 1854.The dichotomy between elementary algebra and modern algebra thenstarted to appear in the subsequent development of logic, with logiciansstrongly divided between the formalistic approach as espoused by Frege,Peano, and Russell, and the algebraic approach followed by C. S. Peirce,Schroeder, and Tarski.

1. Elementary Algebra1.1 Formulas1.2 Laws1.3 Word problems1.4 Cartesian geometry

2. Abstract Algebra2.1 Semigroups2.2 Groups2.3 Rings2.4 Fields2.5 Applications

Algebra


3. Universal Algebra3.1 Concepts3.2 Equational Logic3.3 Birkhoff's Theorem

4. Linear Algebra4.1 Vector Spaces4.2 Associative Algebras

5. Algebraization of mathematics5.1 Algebraic geometry5.2 Algebraic number theory5.3 Algebraic topology5.4 Algebraic logic

6. Free algebras6.1 Free monoids and groups6.2 Free rings6.3 Free combinatorial structures6.4 Free logical structures6.5 Free algebras categorially

BibliographyOther Internet ResourcesRelated Entries

1. Elementary algebra

Elementary algebra deals with numerical terms, namely constants 0, 1,1.5, π, variables x, y, …, and combinations thereof built with operationssuch as +, −, ×, ÷, √, etc. to form such terms as x+1, x × y (standardlyabbreviated xy), x + 3y, and √x.

Terms may be used on their own in formulas such as πr², or in equationsserving as laws such as x+y = y+x, or as constraints such as 2x²−x+3 =

Vaughan Pratt


5x+1 or x²+ y² = 1.

Laws are always true; while they have the same form as constraints theyconstrain only vacuously in that every valuation of their variables is asolution. The constraint x²+y² = 1 has a continuum of solutions forming ashape, in this case a circle of radius 1. The constraint 2x²−x+3 = 5x+1 hastwo solutions, x = 1 or 2, and may be encountered in the solution of wordproblems, or in the determination of the points of intersection of twocurves such as the parabola y = 2x²−x+3 and the line y = 5x+1.

1.1 Formulas

A formula is a term used in the computation of values by hand or machine.Although some attributes of physical objects lend themselves to directmeasurement such as length and mass, others such as area, volume, anddensity do not and must be computed from more readily observed valueswith the help of the appropriate formula. For example the area of arectangle L inches long by W inches wide is given by the formula LW inunits of square inches, the volume of a ball of radius r is 4πr3/3, and thedensity of a solid of mass M and volume V is given by M/V.

Formulas may be combined to give yet more formulas. For example thedensity of a ball of mass M and radius r can be obtained by substitutingthe above formula for the volume of a ball for V in the above formula forthe density of a solid. The resulting formula M/(4πr3/3) is then the desireddensity formula.

1.2 Laws

Laws or identities are equations that hold for all applicable values of theirvariables. For example the commutativity law x+y = y+x holds for all realvalues of x and y. Likewise the associativity law x+(y+z) = (x+y)+z holdsfor all real values of x, y and z. On the other hand, while the law x/(y/z) =

Algebra


zx/y holds for all numerical values of x, it holds only for nonzero values ofy and z in order to avoid the illegal operation of division by zero.

When a law holds for all numerical values of its variables, it also holds forall expression values of those variables. Setting x = M, y = 4πr3, and z = 3in the last law of the preceding paragraph yields M/(4πr3/3) = 3M/(4πr3).The left hand side being our density formula from the preceding section, itfollows from this instance of the above law that its right hand side is anequivalent formula for density in the sense that it gives the same answersas the left hand side. This new density formula replaces one of the twodivisions by a multiplication.

1.3 Word problems

If Xavier will be three times his present age in four years time, how old ishe? We can solve this word problem using algebra by formalizing it as theequation 3x = x + 4 where x is Xavier's present age. The left hand sideexpresses three times Xavier's present age, while the right hand sideexpresses his age in four years' time.

A general rule for solving such equations is that any solution to it is also asolution to the equation obtained by applying some operation to bothsides. In this case we can simplify the equation by subtracting x from bothsides to give 2x = 4, and then dividing both sides by 2 to give x = 2. SoXavier is now two years old.

If Xavier is twice as old as Yvonne and half the square of her age, how oldis each? This is more complicated than the previous example in threerespects: it has more unknowns, more equations, and terms of higherdegree. We may take x for Xavier's age and y for Yvonne's age. The twoconstraints may be formalized as the equations x = 2y and x = y²/2, thelatter being of degree 2 or quadratic.

Vaughan Pratt


Since both right hand sides are equal to x we can infer 2y = y²/2. It istempting to divide both sides by y, but what if y = 0? In fact y = 0 is onesolution, for which x = 2y = 0 as well, corresponding to Xavier andYvonne both being newborns. Setting that solution to one side we can nowlook for solutions in which y is not zero by dividing both sides by y. Thisyields y = 4, in which case x = 2y = 8. So now we have a second solutionin which Xavier is eight years old and Yvonne four.

In the absence of any other information, both solutions are legitimate. Hadthe problem further specified that Yvonne was a toddler, or that Xavierwas older than Yvonne, we could have ruled out the first solution.

1.4 Cartesian geometry

Lines, circles, and other curves in the plane can be expressed algebraicallyusing Cartesian coordinates, named for its inventor Rene Descartes.These are defined with respect to a distinguished point in the plane calledthe origin, denoted O. Each point is specified by how far it is to the rightof and above O, written as a pair of numbers. For example the pair (2.1,3.56) specifies the point 2.1 units to the right of O, measured horizontally,and 3.56 units above it, measured vertically; we call 2.1 the x coordinateand 3.56 the y coordinate of that point. Either coordinate can be negative:the pair (−5, −1) corresponds to the point 5 units to the left of O and 1 unitbelow it. The point O itself is coordinatized as (0, 0).

Lines. Given an equation in variables x and y, a point such as (2, 7) is saidto be a solution to that equation when setting x to 2 and y to 7 makes theequation true. For example the equation y = 3x+5 has as solutions thepoints (0, 5), (1, 8), (2, 11), and so on. Other solutions include (.5, 6.5),(1.5, 9.5), and so on. The set of all solutions constitutes the unique straightline passing through (0, 5) and (1, 8). We then call y = 3x+5 the equationof that line.

Algebra


Circles. By Pythagoras's Theorem the square of the distance between twopoints (x, y) and (xʹ′, yʹ′) is given by (xʹ′−x)²+(yʹ′−y)². As a special case ofthis, the square of the distance of the point (x, y) to the origin is x²+y². Itfollows that those point at distance r from the origin are the solutions in xand y to the equation x²+y² = r². But these points are exactly those formingthe circle of radius r centered on O. We identify this equation with thiscircle.

Varieties The roots of any polynomial in x and y form a curve in the planecalled a one-dimensional variety of degree that of the polynomial. Thuslines are of degree 1, being expressed as polynomials ax+by+c, whilecircles centered on (xʹ′, yʹ′) are of degree 2, being expressed as polynomials(x−xʹ′)²+(y−yʹ′)²−r². Some varieties may contain no points, for example x²+y²+1, while others may contain one point, for example x²+y² having theorigin as its one root. In general however a two-dimensional variety willbe a curve. Such a curve may cross itself, or have a cusp, or even separateinto two or more components not connected to each other.

Space The two-dimensional plane is generalized to three-dimensionalspace by adding to the variables x and y a third variable z corresponding tothe third dimension. The conventional orientation takes the first dimensionto run from east to west, the second from south to north, and the third frombelow to above. Points are then triples, for example the point (2, 5, −3) is2 units to the east of the origin, 5 units to the north of it, and 3 units belowit.

Planes and spheres. These are the counterparts in space of lines andcircles in the plane. An equation such as z = 3x + 2y defines not a straightline but rather a flat plane, in this case the unique plane passing throughthe points (0, 1, 2), (1, 0, 3), and (1, 1, 5). And the sphere of radius rcentered on the origin is given by x²+y²+z² = r². The roots of a polynomialin x, y and z form a surface in space called a two-dimensional variety, of

Vaughan Pratt


degree that of the polynomial, just as for one-dimensional varieties. Thusplanes are of degree 1 and spheres of degree 2.

These methods generalize to yet higher dimensions by adding yet morevariables. Although the geometric space we experience physically islimited to three dimensions, conceptually there is no limit to the number ofdimensions of abstract mathematical space. Just as a line is a one-dimensional subspace of the two-dimensional plane, and a plane is a two-dimensional subspace of three-dimensional space, each specifiable with anequation, so is a hyperplane a three-dimensional subspace of four-dimensional space, also specifiable with an equation such as w = 2x − 7y +z.

2. Abstract Algebra

Elementary algebra fixes some domain, typically the reals or complexnumbers, and works with the equations holding within that domain.Abstract or modern algebra reverses this picture by fixing some set A ofequations and studying those domains for which those equations areidentities. For example if we take the set of all identities expressible withthe operations of addition, subtraction, and multiplication and constants 0and 1 that hold for the integers, then the algebras in which those equationshold identically are exactly the commutative rings with identity.

Historically the term modern algebra came from the title of the first threeeditions of van der Waerden's classic text of that name, renamed simply"Algebra" for its fourth edition in 1955. Volume 1 treated groups, rings,general fields, vector spaces, well orderings, and real fields, while Volume2 considered mainly linear algebra, algebras (as vector spaces with acompatible multiplication), representation theory, ideal theory, integralalgebraic elements, algebraic functions, and topological algebra. On theone hand modern algebra has since gone far beyond this curriculum, on

Algebra


the other this considerable body of material is already more than what canbe assumed as common knowledge among graduating Ph.D. students inmathematics, for whom the typical program is too short to permitmastering all this material in parallel with focusing on their area ofspecialization.

A core feature of abstract algebra is the existence of domains wherefamiliar laws fail to hold. A striking example is commutativity ofmultiplication, which as we noted in the introduction need not hold for themultiplication of an arbitrary group, even so simple a group as the sixpermutations of three letters.

2.1 Semigroups

We begin with the concept of a binary operation on a set X, namely afunction f: X² → X such that f(x, y) is an element of X for all elements x, yof X. Such an operation is said to be associative when it satisfies f(f(x, y),z) = f(x, f(y, z)) for all x, y, z in X.

A semigroup is a set together with an associative operation, called themultiplication of the semigroup and notated xy rather than f(x, y).

The product xx of an element with itself is denoted x². Likewise xxx isdenoted x³ and so on.

Examples

The set of all nonempty words over a given alphabet underthe operation of concatenation.The set of all functions f: X → X on a set X under theoperation of function composition.The set of integer n × n matrices under matrix multiplication,for a fixed positive integer n.

Vaughan Pratt


Concatenation uv of words u, v is associative because when a word is cutinto two, the concatenation of the two parts is the original word regardlessof where the cut is made. The concatenation of al and gebra is the same asthat of algeb and ra, illustrating associativity of concatenation for the casex = al, y = geb, z = ra.

Composition f·g of functions f, g is associative via the reasoning

for all x in X, whence f·(g·h) = (f·g)·h.

A semigroup H is a subsemigroup of a semigroup G when H is a subset ofG and the multiplication of G restricted to H coincides with that of H.Equivalently a subsemigroup of G is a subset H of G such that for all x, yin H, xy is in H.

A binary operation is called commutative when it satisfies f(x, y) = f(y, x)for all x, y in X. A commutative semigroup is a semigroup whose operation

(f·(g·h))(x) = f((g·h)(x))= f(g(h(x)))= (f·g)(h(x))= ((f·g)·h)(x)

Examples

The semigroup of all nonempty words over a given alphabethas as a subsemigroup the words of even length; however thewords of odd length do not form a subsemigroup because theconcatenation of two odd-length words is not of odd length.The semigrop of all functions f: X → X on a set X underfunction composition has as subsemigroups the injective orone-to-one functions, the surjective or onto functions, and thebijections or permutations.

Algebra


is commutative. All the examples so far have been of noncommutativesemigroups. The following illustrate the commutative case.

An element x of X is a left identity for f when f(x, y) = y for all y in X, and aright identity when f(y, x) = y for all y in X. An identity for f is an elementthat is both a left identity and a right identity for f. An operation f can haveonly one identity, because when x and y are identities they are both equalto f(x,y).

A monoid is a semigroup containing an identity for the multiplication ofthe semigroup, notated 1.

Examples

The set of positive integers under addition.The set of all integers under addition.The set of words on a one-letter alphabet under concatenation.The set {0, 1} of bits (binary digits) under any of theoperations AND, OR, XOR.The set 2X of subsets of a fixed set X under any of the settheoretic operations intersection, union, symmetric difference.The set of vectors in the upper right quadrant of the planeunder vector addition.The same but omitting the origin.The set of all three-dimensional vectors under vector addition.The set of polynomials in one variable x with integercoefficients under polynomial addition.The set of integer m × n matrices under matrix addition, forfixed positive integers m,n.

Examples

The identity for concatenation is the empty word. Hence

Vaughan Pratt


A monoid H is a submonoid of a monoid G when it is a subsemigroup ofG that includes the identity of G.

2.2 Groups

When two elements x, y of a monoid satisfy xy = 1 we say that x is the leftinverse of y and y is the right inverse of x. An element y that is both a leftand right inverse of x is called simply an inverse of x.

A group is a monoid every element of which has an inverse.

The cardinality of a group is traditionally referred to as its order. A groupelement g is said to be of order n when n is the least positive integer forwhich gn = 1.

words under concatenation form a monoid when the emptyword is allowed.The identity for addition is zero, or the origin in the case ofvector addition. Hence any of the above examples ofsemigroups for which the operation is addition forms amonoid if and only if it contains zero.The identity for composition is the identity function 1X: X →X defined as 1X(x) = x for all x in X, whence the semigroup ofall functions on a set X forms a monoid.

Examples

The monoid of integers under addition, because every integerx has inverse −x.The set {0, 1} of bits (binary digits) under the operation XOR,because each bit is its own inverse: 0 XOR 0 = 1 XOR 1 = 0.The monoid SX of bijections or permutations f: X → X undercomposition, because every permutation has an inverse f−1.

Algebra


A subgroup of a group G is a submonoid of G closed under inverses. Themonoids of natural numbers and of even integers are both submonoids ofthe monoid of integers under addition, but only the latter submonoid is asubgroup, being closed under negation, unlike the natural numbers.

An abelian group is a group whose operation is commutative. The groupoperation of an abelian group is conventionally referred to as additionrather than multiplication, and abelian groups are sometimes calledadditive groups.

A cyclic group is a group G with an element g such that every element ofG is of the form gi for some positive integer i. Cyclic groups are abelianbecause gigj = gi+j = gj gi. The group of integers under addition, and thegroups of integers mod n for any positive integer n, all form cyclic groups,with 1 as a generator in every case. All cyclic groups are isomorphic toone of these. There are always other generators when the group is of order3 or more, for example −1, and for groups of prime order every nonzeroelement is a generator.

When X has n elements Sn is of order n!. Sn is abelian if andonly if n ≤ 2.The monoid of rotations of the plane about a point undercomposition, because every rotation can be reversed. Thisgroup is called the circle group, denoted SO(2).The monoid of rotations of three-dimensional space about apoint under composition, again because every rotation can bereversed. This group is denoted SO(3).The monoid of symmetries (rotations and reflections) of theregular n-gon about its center that carry the n-gon into itself,again by reversibility. This group is called the dihedral groupDn, and is of order 2n. Like Sn, Dn is abelian if and only if n ≤2; in particular D3 = S3.

Vaughan Pratt


2.3 Rings

A ring is an abelian group that is also a monoid by virtue of having asecond operation, called the multiplication of the ring. Zero annihilates,meaning that 0x = x0 = 0. Furthermore multiplication distributes overaddition (the group operation) in both arguments. That is, x(y+z) = xy + xzand (x+y)z = xz + yz.

In all but the last example the integers (other than the integer n giving thesize of the matrices) may be replaced by any of the rationals, the reals, orthe complex numbers. When this is done with the reals the third exampledegenerates to just the reals because √2 is real (and likewise with thecomplex numbers), but with the rational numbers it does not degeneratebecause √2 is irrational.

2.4 Fields

A field is a ring for which the multiplicative monoid of nonzero ringelements is an abelian group. That is, multiplication must be commutative,and every nonzero element x must have a reciprocal 1/x.

Examples

The group of integers under integer multiplication.The group of polynomials in one variable x with integercoefficients under polynomial multiplication.The group of integer n × n matrices under matrixmultiplication, for a fixed positive integer n.The group of numbers of the form a+b√2 where a and b areinteger, because (a+b√2)(c+d√2) = ac+2bd+(bc+ad)√2.The group of integers mod n for n ≥ 2, because (x+an)(y+bn)= xy + (xb + ya + abn)n, showing that ordinary integermultiplication is compatible with congruence mod n.

Algebra


The last example does not generalize directly to other moduli. Howeverfor any modulus that is a power pn of a prime, it can be shown that thereexists a unique multiplication making the group Zpn a ring in a way thatmakes the nonzero elements of the ring a cyclic (and therefore abelian)group under the multiplication, and hence making the ring a field. Thefields constructed in this way are the only finite fields.

2.5 Applications

Why study entire classes? Well, consider for example the set Z of integersalong with the binary operation of addition x+y, the unary operation ofnegation −x, and the constant 0. These operations and the constant satisfyvarious laws such as x+(y+z) = (x+y)+z, x+y = y+x, x+0 = x, and x+(−x) =0. Now consider any other algebra with operations that not only have thesame names but also satisfy the same laws (and possibly more), called amodel of those laws. Such an algebra could serve any of the following

Examples

The ring of rationals, because rational multiplication iscommutative and every nonzero rational m/n has reciprocaln/m.The ring of reals, and the ring of complex numbers, for theanalogous reasons.The ring of numbers of the form a+b√2 where a and b arerational, because a+b√2 is zero only when a=b=0, andotherwise has reciprocal (a−b√2)/(a²−2b²); called the field ofquadratic irrationals.The ring Zp of integers modulo a prime p, because themultiplicative monoid of nonzero numbers includes a numberg such that gp−1 = 1 and every nonzero number has the formgi for some integer i, and hence has an inverse, namely gp−1−i.

Vaughan Pratt


purposes.

(i) It could tell us to what extent the equational laws holding of theintegers characterize the integers. Since the set {0, 1} of integers mod 2under addition and negation satisfies all the laws that the integers do, weimmediately see that no single equational property of the integers tells usthat there are infinitely many integers. On the other hand any finite modelof the equational theory of the integers necessarily satisfies some law thatthe integers don't satisfy, in particular the law x+x+…+x = 0 where thenumber of xs on the left hand side is the size of the model. Since theequational theory of the integers contains no such law we can tell from itstheory as a whole that the integers must be an infinite set. On the otherhand the rational numbers under addition and negation satisfy exactly thesame equational properties as the integers, so this theory does notcharacterize the algebra of integers under addition and subtraction withsufficient precision to distinguish it from the rationals.

(ii) It could provide us with a useful new domain that can be substitutedfor the integers in any application depending only on equational propertiesof the integers, but which differs from the integers in other (necessarilynonequational) useful respects. For example the rationals, which satisfythe same laws as we just noted, differ in having the density property, thatbetween any two rationals there lies another rational. Another difference isthat it supports division: whereas the ratio of two integers is usually not aninteger, the ratio of two rationals is always a rational. The reals also satisfythe same equations, and like the rationals are dense and support division.Unlike the rationals however the reals have the completeness property,that the set of all upper bounds of any nonempty set of reals is eitherempty or has a least member, needed for convergent sequences to have alimit to converge to.

This idea extends to other operations such as multiplication and division,

Algebra


as with fields. A particularly useful case of such a generalization is givenby the use of complex numbers in Cartesian geometry. When x and yrange over the field of reals, x²+y²=1 describes the ordinary Euclideancircle in two dimensions, but when the variables range over the complexnumbers this equation describes the complex counterpart of the circle,visualizable as a two-dimensional surface embedded in four realdimensions (regarding the complex plane as having two real dimensions).Or if the variables range over the integers mod 7, which form a field underthe usual arithmetic operations mod 7, the circle consists of eight points,namely (±1, 0), (0, ±1), and (±2, ±2). Certain theorems about theEuclidean circle provable purely algebraically remain provable about theseother kinds of circles because all the equations on which the proof dependscontinue to hold in these other fields, for example the theorem that a lineintersects a circle in at most two points.

(iii) It could help us decide whether some list of equational laws intendedto axiomatize the integers is complete in the sense that any equationholding of the integers follows from the laws in that list. If some structuresatisfies all the axioms in the list, but not some other equation that holds ofthe integers, then we have a witness to the incompleteness of theaxiomatization. If on the other hand we can show how to construct anyalgebra satisfying the axioms from the algebra of integers, limitingourselves only to certain algebraic constructions, then by a theorem ofBirkhoff applicable to those constructions we can infer that theaxiomatization is complete.

(iv) It could give another of way of defining a class, besides the standardway of listing axioms. In the case at hand, the class of all algebras with aconstant, a unary operation, and a binary operation, satisfying all the lawssatisfied by the integers, is exactly the class of abelian groups.

A handy contraction for "model of the theory of" that has been proposed

Vaughan Pratt


from time to time without ever getting serious traction is "homologue of".The above says that an abelian group is any homologue of the group ofintegers. Similarly a group can be defined as any homologue of the groupof rotations of the sphere, or of the nonsingular 2 × 2 real matrices undermatrix multiplication (equivalently the group of automorphisms of theplane). A commutative ring can be defined as any homologue of the ringof integers. A ring is any homologue of the ring of 2 × 2 integer matrices,or equivalently real matrices.

3. Universal Algebra

Universal algebra is the next level of abstraction after abstract algebra.Whereas elementary algebra treats equational reasoning in a particularalgebra such as the field of reals or the field of complex numbers, andabstract algebra studies particular classes of algebras such as groups, rings,or fields, universal algebra studies classes of classes of algebras. Much asabstract algebra numbers groups, rings, and fields among its basic classes,so does universal algebra count varieties, quasivarieties, and elementaryclasses among its basic classes of classes.

A model of a theory is a structure for which all the equations of that theoryare identities. Terms are built up from variables and constants using theoperations of the theory. An equation is a pair of terms; it is satisfied by analgebra when the two terms are equal under all evaluations of the nvariables appearing in the terms, equivalently when they denote the samen-ary operation. A quasiequation is a pair consisting of a finite set ofequations, the premises, and another equation, the conclusion; it issatisfied by an algebra when the two terms of the conclusion are equalunder all evaluations of the n variables appearing in the terms satisfyingthe premises. A first order formula is a quantified Boolean combination ofrelational terms.

Algebra


A variety is the class of all models of a set of equations. A quasivariety isthe class of all models of a set of quasiequations. An elementary class s isthe class of all models of a set of first-order formulas.

Quasivarieties have received much less attention than either varieties orelementary classes, and we accordingly say little about them here.Elementary classes are treated in sufficient depth elsewhere in thisencyclopedia that we need not consider them here. We therefore focus inthis section on varieties.

Abelian groups, groups, rings, and vector spaces over a given field allform varieties.

A central result in this area is the theorem that a lattice arises as the latticeof subalgebras of some algebra if and only if it arises as the lattice ofcongruences on some algebra. Lattices of this sort are called algebraiclattices. When the congruences of an algebra permute, its congruencelattice is modular, a strong condition facilitating the analysis of finitealgebras in particular.

3.1 Concepts

Familiar theorems of number theory emerge in algebraic form foralgebras. An algebra A is called directly irreducible or simple when itslattice of congruences is the two-element lattice consisting of A and theone-element algebra, paralleling the notion of prime number p as a numberwhose lattice of divisors has two elements p and 1. However thecounterpart of the fundamental theorem of arithmetic, that every positiveinteger factors uniquely as a product of primes, requires a more delicatekind of product than direct product. Birkhoff's notion of subdirect productenabled him to prove the Subdirect Representation Theorem, that everyalgebra arises as the subdirect product of its subdirectly irreduciblequotients. Whereas there are many subdirectly irreducible groups, the only

Vaughan Pratt


subdirectly irreducible Boolean algebra is the initial or two-element one,while the subdirectly irreducible rings satisfying xn = x for some n > 1 areexactly the finite fields.

Another central topic is duality: Boolean algebras are dual to Stone spaces,complete atomic Boolean algebras are dual to sets, distributive latticeswith top and bottom are dual to partially ordered sets, algebraic lattices aredual to semilattices, and so on. Duality provides two ways of looking at analgebra, one of which may turn out to be more insightful or easier to workwith than the other depending on the application.

The structure of varieties as classes of all models of some equationaltheory is also of great interest. The earliest result in this area is Birkhoff'stheorem that a class of algebras is a variety if and only if it is closed underformation of quotients (homomorphic images), subalgebras, and arbitrary(including empty and infinite) direct products. This "modern algebra"result constitutes a completeness theorem for equational logic in terms ofits models. Its elementary counterpart is the theorem that the equationaltheories on a free algebra F(V), defined as the deductively closed sets ofequations that use variables from V, are exactly its substitutivecongruences.

A locally finite variety is one whose finitely generated free algebras arefinite, such as pointed sets, graphs (whether of the directed or undirectedvariety), and distributive lattices. A congruence permutable variety is avariety all of whose algebras are congruence permutable. Maltsevcharacterized these in terms of a necessary and sufficient condition ontheir theories, namely that F(3) contain an operation t(x, y, z) for whicht(x, x, y) = t(y, x, x) = y are in the theory. Analogous notions arecongruence distributivity and congruence modularity, for which there existanalogous syntactic characterizations of varieties of algebras with theseproperties. A more recently developed power tool for this area is

Algebra


McKenzie's notion of tame congruences, facilitating the study of thestructure of finite algebras.

Within the algebraic school, varieties have been defined with theunderstanding that the operations of a signature form a set. Insights fromcategory theory, in particular the expression of a variety as a monad,defined as a monoid object in the category CC of endofunctors of acategory C (Set in the case of ordinary universal algebra) indicate that acleaner and more general notion of variety is obtained when the operationscan form a proper class. For example the important classes of completesemilattices, CSLat, and complete atomic Boolean algebras, CABA, formvarieties only with this broader notion of signature. In the narrowalgebraic sense of variety, the dual of a variety can never be a variety,whereas in the broader monadic notion of variety, the variety Set of sets isdual to CABA while CSLat is self-dual.

3.2 Equational Logic

Axiom systems. Identities can also be used to transform equations toequivalent equations. When those equations are themselves identities forsome domain, the equations they are transformed into remain identities forthat domain. One can therefore start from some finite set of identities andmanufacture an unlimited number of new identities from them.

For example if we start from just the two identities (x+y)+z = x+(y+z) andx+y = y+x, we can obtain the identity (w+x)+(y+z) = (w+y)+(x+z) via thefollowing series of transformations. (w+x)+(y+z) = ((w+x)+y)+z = (w+(x+y))+z = (w+(y+x))+z = ((w+y)+x)+z = (w+y)+(x+z).

This process of manufacturing new identities from old is called deduction.Any identity that can be generated by deduction starting from a given setA of identities is called a consequence of A. The set of all consequences ofA is called the deductive closure of A. We refer to A as an axiomatization

Vaughan Pratt


of its deductive closure. A set that is its own deductive closure is said to bedeductively closed. It is straightforward to show that a set is deductivelyclosed if and only if it is the deductive closure of some set.

An equational theory is a deductively closed set of equations, equivalentlythe set of all consequences of some set A of equations. Every theoryalways has itself as its own axiomatization, but it will usually also havesmaller axiomatizations. A theory that has a finite axiomatization is said tobe finitely based or finitely axiomatizable.

Effectiveness. Finitely based theories can be effectively enumerated. Thatis, given a finite set A of equations, one can write a computer program thatprints consequences of A for ever in such a way that every consequence ofA will appear at some finite position in the infinite list of all consequences.The same conclusion obtains when we weaken the requirement that A befinite to merely that it can be effectively enumerated. That is, if theaxiomatization is effectively enumerable so is its deductive closure.

(In reconciling the finite with the infinite, bear in mind that if we list allthe natural numbers 0, 1, 2, … in order, we obtain an infinite list everymember of which is only finitely far from the beginning, and also has awell-defined predecessor and successor. Only if we attempt to pad this listout at the "end" with infinite numbers does this principle break down. Oneway to visualize there being an "end" that could have more elementsbeyond it is to consider the rationals of the form 1/n for all nonzerointegers n, ordered backwards. This list starts out 1/1, 1/2, 1/3, … and afterlisting infinitely many positive rationals switches over to the negativerationals, with no first such, finally ending with −1/3, −1/2, −1/1. Theentire list is discrete in the sense that every rational except the endpoints1/1 and −1/1 has a well-defined predecessor and successor in this subset ofthe rationals, unlike the situation for the set of all rationals. This would nolonger be the case were we to introduce the rational 0, which would have

Algebra


neither a predecessor nor a successor.)

Equational Logic. Our informal account of deduction can be formalizedin terms of five rules for producing new identities from old. In thefollowing, s and t denote arbitrary terms.

"Consistently" in this context means that if a term is substituted for oneoccurrence of a given variable, the same term must be substituted for alloccurrences of that variable in both s and t. We could not for exampleappeal to R5 to justify substituting u+v for x in the left hand side of x+y =y+x and v+u for x in the right hand side, though some other rule mightpermit it.

An equational theory as a set of pairs of terms amounts to a binary relationon the set of all terms. Rules R1–R3 correspond to respectively reflexivity,symmetry, and transitivity of this binary relation, i.e. these three rulesassert that an equational theory is an equivalence relation. Rule R4expresses the further property that this binary relation is a congruence.Rule R5 further asserts that the relation is a substitutive congruence. It canbe shown that a binary relation on the set of terms is an equational theoryif and only if it is a substitutive congruence. These five rules thereforecompletely axiomatize equational logic in the sense that everyconsequence of a set A of equations can be produced from A via finitely

R1. From nothing infer t = t.R2. From s = t infer t = s.R3. From s = t and t = u infer s = uR4. From s1 = t1, s2 = t2, …, sn = tn infer f(s1, s2, …, sn)= f(t1, t2,

…, tn), where f is an n-ary operation.R5. From s = t infer sʹ′ = tʹ′ where sʹ′ and tʹ′ are the terms resulting

from consistently substituting terms for variables in s and trespectively.

Vaughan Pratt


many applications of these five rules.

3.3 Birkhoff's Theorem

A variety is by definition the class of models of some equational theory. In1935 Birkhoff provided an equivalent characterization of varieties as anyclass closed under quotients (homomorphic images), direct products, andsubalgebras. These notions are defined as follows.

Given two algebras (X, f1, … fk) and (Y, g1, … gk), a homomorphism h:(X, f1, … fk) → (Y, g1, … gk) is a function h: X → Y satisfying h(fi(x0, …,xni−1)) = gi(h(x0), …, h(xni−1))) for each i from 1 to k where ni is the arityof both fi and gi.

A subalgebra of an algebra is a set of elements of the algebra closed underthe operations of the algebra.

Let I be an arbitrary set, which may be empty, finite, or infinite. A family<Ai>i∈I of algebras (Xi, f1i, …fki) indexed by I consists of one algebra Aifor each element i of I. We define the direct product ΠAi (or Πi∈IAi infull) of such a family as follows.

The underlying set of ΠAi is the cartesian product ΠXi of the underlyingsets Xi, and consists of those I-tuples whose i-th element is some elementof Xi. (I may even be uncountable, but in this case the nonemptiness of Xias a consequence of the nonemptiness of the individual Xi's is equivalentto the axiom of choice. This should be kept in mind for any constructiveapplications of Birkhoff's theorem.)

The j-th operation of ΠAi, of arity nj, takes an nj-tuple t of elements ofΠXi and produces the I-tuple <fji( t1i, … tnj

i)>i∈I where tki is the i-thcomponent of the k-th component of t for k from 1 to nj.

Algebra


Given two algebras A, B and a homomorphism h: A → B, thehomomorphic image h(A) is the subalgebra of B consisting of elements ofthe form h(a) for a in A.

Given a class C of algebras, we write P(C) for the class of all algebrasformed as direct products of families of algebras of C, S(C) for the class ofall subalgebras of algebras of C, and H(C) for the class of allhomomorphic images of algebras of C.

It is relatively straightforward to show that any equation satisfied by allthe members of C is also satisfied by all the members of P(C), S(C), andH(C). Hence for a variety V, P(V) = S(V) = H(V).

Birkhoff's theorem is the converse: for any class C such that P(C) = S(C) =H(C), C is a variety. In fact the theorem is slightly stronger: for any classC, HSP(C) is a variety. That is, to construct all the models of the theory ofC it suffices to close C first under direct products, then under subalgebras,and finally under homomorphic images; that is, later closures do notcompromise earlier ones provided P, S, and H are performed in that order.

A basic application of Birkhoff's theorem is in proving the completenessof a proposed axiomatization of a class C. Given an arbitrary model of theaxioms, it suffices to show that the model can be constructed as thehomomorphic image of a subalgebra of a direct product of algebras of C.

This completeness technique complements the completeness observed inthe previous section for the rules of equational logic.

4. Linear Algebra

4.1 Vector Spaces

Sibling to groups, rings, and fields is the class of vector spaces over any

Vaughan Pratt


given field, constituting the universes of linear algebra. Vector spaces lendthemselves to two opposite approaches: axiomatic or abstract, andsynthetic or concrete. The axiomatic approach takes fields (whence rings,whence groups) as a prerequisite; it first defines a notion of R-module asan abelian group with a scalar multiplication over a given ring R, and thendefines a vector space to be an R-module for which R is a field. Thesynthetic approach proceeds via the familiar representation of vectorspaces over the reals as n-tuples of reals, and of linear transformationsfrom m-dimensional to n-dimensional vector spaces as m × n matrices ofreals. For the full generality of vector spaces including those of infinitedimension, n need not be limited to finite numbers but can be any cardinal.

The abstract approach, as adopted by such classical texts as Mac Lane andBirkhoff, has a certain purist appeal and is ideally suited to mathematicsmajors. The concrete approach has the benefit of being able to substitutecalculus or less for groups-rings-fields as a prerequisite, suiting it toservice courses for scientists and engineers needing only finite-dimensional matrix algebra, which enjoys enormous practicalapplicability. Linear algebra over other fields, in particular finite fields, isused in coding theory, quantum computing, etc., for which the abstractapproach tends to be better suited.

For any field F, up to isomorphism there is exactly one vector space overF of any given finite dimension. This is a theorem in the abstractapproach, but is an immediate consequence of the representation in theconcrete approach (the theorem is used in relating the two approaches).

Another immediate consequence of the concrete approach is duality forfinite-dimensional vector spaces over F. To every vector space V, of anydimension, corresponds its dual space V* comprised of the functionals onV, defined as the linear transformations f : V→F, viewing the field F as theone-dimensional vector space. The functionals form a vector space under

Algebra


coordinatewise addition (f+g)(u) = f(u)+g(u) and multiplication (xgf)(u) =x(f(u)) by any scalar x in F, and we take V* to be that space. This operationon vector spaces extends to the linear transformations f : U→V as f*:V*→U* defined such that f maps each functional g: V→F to ggf : U→F.Repeating this operation produces a vector space that, in the finite-dimensional case, is isomorphic to V, that is, V ≅ V**, making theoperation an involution. The essence of duality for finite-dimensionalvector spaces resides in its involutary nature along with the reversal of thelinear transformations.

This duality is easily visualized in the concrete approach by viewing lineartransformations from U to V as m × n matrices. The duality simplytransposes the matrices while leaving the machinery of matrixmultiplication itself unchanged. It is then immediate that this operation isan involution that reverses maps—the m × n matrix linearly transformingan n-dimensional space U to an m-dimensional one V transposes to an n × m matrix linearly transforming the m-dimensional space V* to the n-dimensional space U*.

4.2 Associative Algebras

The linear transformations f : V→V on a vector space V can be added,subtracted, and multiplied by scalars, pointwise in each case, and henceform a vector space. When the space has finite dimension n, the lineartransformations are representable as n × n matrices.

In addition they can be multiplied, whence they form a vector spaceequipped with a bilinear associative operation. In the finite-dimensionalcase, multiplication is realized as the usual matrix product. Vector spacesfurnished with such a product constitute associative algebras. Up toisomorphism, all associative algebras arise in this way whether of finite orinfinite dimension, providing a satisfactory and insightful characterization

Vaughan Pratt


of the notion in lieu of an axiomatic characterization, not given here.

Well-known examples of associative algebras are the reals, the complexnumbers, and the quaternions. Unlike vector spaces, many nonisomorphicassociative algebras of any given dimension greater than one are possible.

A class of associative algebras of interest to physicists is that of theClifford algebras. Clifford algebras over the reals (which as vector spacesare Euclidean spaces) generalize complex numbers and quaternions bypermitting any number of formal quantities e analogous to i = √−1 to beadjoined to the field of reals. The common feature of these quantities isthat each satisfies either e² = −1 or e² = 1. Whereas there are a great manyassociative algebras of low dimension, only a few of them arise as Cliffordalgebras. The reals form the only one-dimensional Clifford algebra, whilethe hyperbolic plane, defined by e² = 1, and the complex plane, defined bye² = −1, are the two two-dimensional Clifford algebras. The hyperbolicplane is just the direct square of the real field, meaning that its product iscoordinatewise, (a, b)(c, d) = (agc, bgd), unlike that of the complex planewhere it defined by (a, b)(c, d) = (agc−bd, ad+bc). The two four-dimensional Clifford algebras are the 2 × 2 matrices and the quaternions.Whereas the 2 × 2 matrices contain zero divisors (nonzero matrices whoseproduct is zero), and so form only a ring, the quaternions contain no zerodivisors and so form a division ring. Unlike the complex numbers howeverthe quaternions do not form a field because their multiplication is notcommutative. Complex multiplication however makes the complex planea commutative division ring, that is, a field.

5. Algebraization of mathematics

A number of branches of mathematics have benefited from the perspectiveof algebra. Each of algebraic geometry and algebraic combinatorics has anentire journal devoted to it, while algebraic topology, algebraic logic, and

Algebra


algebraic number theory all have strong followings. Many other morespecialized areas of mathematics have similarly benefited.

5.1 Algebraic geometry

Algebraic geometry begins with what we referred to in the introduction asshapes, for example lines y = ax+b, circles x²+y² = r², spheres x²+y²+z² =r², conic sections f(x, y) = 0 where f is a quadratic polynomial in x and y,quadric surfaces f(x, y, z) = 0 with f again quadratic, and so on.

It is convenient to collect the two sides of these equations on the left sothat the right side is always zero. We may then define a shape or variety toconsist of the roots of a polynomial.

Ordinary analytical or Cartesian geometry is conducted over the reals.Algebraic geometry is more commonly conducted over the complexnumbers, or more generally over any algebraically closed field. Thevarieties definable in this way are called affine varieties.

Sometimes however algebraic closure is not desirable, for example whenworking at the boundary of algebraic geometry and number theory wherethe field may be finite, or the rationals.

Many kinds of objects are characterized by what structure their maps holdinvariant. Posets transform via monotone functions, leaving orderinvariant. Algebras transform via homomorphisms, leaving the algebraicstructure invariant. In algebraic geometry varieties transform via regularn-ary functions f: An → A, defined as functions that are locally rationalpolynomials in n variables. Locally rational means that at each point of thedomain of f there exists a neighborhood on which f is the ratio of twopolynomials, the denominator of which is nonzero in that neighborhood.

This notion generalizes to regular functions f: An → An defined as m-

Vaughan Pratt


tuples of regular n-ary functions.

Given two varieties V, Vʹ′ in An and Am respectively, a regular functionfrom An to Am whose restriction to V is a function from V to Vʹ′ is called aregular function of varieties. The category of affine varieties is thendefined to have as its objects all affine varieties and as its morphisms allregular functions thereof.

Polynomials being continuous, one would expect regular functionsbetween varieties to be continuous also. A difficulty arises with the shapesof varieties, where there can be cusps, crossings, and other symptoms ofsingularity. What is needed here is a suitable topology by which to judgecontinuity.

The trick is to work not in affine space but its projective space. Toillustrate with Euclidean three-space, its associated projective space is theunit sphere with antipodal points identified, forming a two-dimensionalmanifold. Equivalently this is the space of all (unoriented) lines throughthe origin. Given an arbitrary affine space, its associated projective spaceis the space of all such lines, understood as a manifold.

The topology on projective space appropriate for algebraic geometry is theZariski topology, defined not by its open sets but rather by its closed sets,which are taken to be the algebraic sets, namely those sets constituting thecommon zeros of a set of homogeneous polynomials. The crucial theoremis then that regular maps between affine varieties are continuous withrespect to the Zariski topology.

5.2 Algebraic number theory

Algebraic number theory has adopted these generalizations of algebraicgeometry. One class of varieties in particular that has been of greatimportance to number are elliptical curves.

Algebra


A celebrated success of algebraic number theory has been Andrew Wiles'proof of Fermat's so-called "last theorem." This had remained an openproblem for over three and a half centuries.

5.3 Algebraic topology

Algebraic topology analyzes the holes and obstructions in connectedtopological spaces. A topologist is someone who imagines all objects to bemade of unbreakable but very pliable playdough, and therefore does notsee the need to distinguish between a coffee cup and doughnut becauseeither can be turned into the other. Topology is concerned with thesimilarities and differences between coffee cups with n handles, surfaceswith n holes, and more complicated shapes. Algebraic topology expressesthe invariants of such shapes in terms of their homotopy groups andhomology groups.

5.4 Algebraic logic

Algebraic logic got off to an early start with Boole's introduction ofBoolean algebra in an 1847 pamphlet. The methods of modern algebrabegan to be applied to Boolean algebra in the 20th century. Algebraiclogic then broadened its interests to first order logic and modal logic.Central algebraic notions in first order logic are ultraproducts, elementaryequivalence, and elementary and pseudoelementary varieties. Tarski'scylindric algebras constitute a particular abstract formulation of first orderlogic in terms of diagonal relations coding equality and substitutionrelations encoding variables. Modal logic as a fragment of first order logicis made algebraic via Boolean modules.

6. Free Algebras

Given any system such as integer arithmetic or real arithmetic, we can

Vaughan Pratt


write T for the set of all definite terms, constituting the definite language,and T[V] for the larger indefinite language permitting variables drawnfrom a set V in place of some of the constant symbols. When V containsonly a single variable "x", T[{"x"}] is usually abbreviated to T["x"] or justT[x] which is usually unambiguous. This convention extends to thealgebra Φ of terms of T together with its list of operation symbols view asoperations for combining terms; we write Φ[V] and call it the free algebraon V. The initial algebra Φ is Φ[∅], the case of no variables.

We now explore free algebras in more depth.

By a C-algebra we shall mean a member of a class C of algebras. Forexample a Boolean algebra is a member of the class of all Booleanalgebras. A free C-algebra is an algebra that lives at the frontier of syntaxand semantics. On the one hand it is semantic by virtue of being a memberof C. On the other it is syntactic in that its elements behave like terms.

Free algebras can be approached from either side. From the syntactic side,the free C-algebra B on a set X arises as a quotient of the term algebraformed from X (viewed as a set of variables) using the operation symbolsand constants common to the algebras of C. The quotient identifies thoseterms that have the same value for all algebras A of C and all valuationsassigning values in A to the variables of X. This performs just enoughidentifications to satisfy every law of C (thereby making this quotient a C-algebra) while still retaining the syntactic essence of the original termalgebra in a sense made more precise by the following paragraph.

(Since the concept of a term algebra can seem a little circular in places, amore detailed account may clarify the concept. Given the language of C,meaning the operation symbols and constant symbols common to thealgebras of C, along with a set X of variables, we first form the underlyingset of the algebra, and then interpret the symbols of the language as

Algebra


operations on and values in that set. The set itself consists of the termsbuilt in the usual way from those variables and constant symbols using theoperation symbols; in that sense these elements are syntactic. But now wechange our point of view by treating those elements as semantic, and welook to the constant symbols and operation symbols of the language assyntactic entities needing to be interpreted in this semantic domain (albeitof terms) in order to turn this set of terms into an algebra of terms. Weinterpret each constant symbol as itself. And we interpret each n-aryoperation symbol f as the n-ary operation that takes any n terms t1, …, tnas its n arguments and returns the single term f(t1, …, tn). Note that thisinterpretation of f only returns a term, it does not actually build it. All termbuilding was completed when we produced the underlying set of thealgebra.)

From the semantic side, a C-algebra B together with a subset X of Bthought of as variables is said to be a free C-algebra on X, or is freelygenerated by X, when, given any C-algebra A, any valuation in A of thevariables in X (that is, any function f : X→A) uniquely extends to ahomomorphism h: B→A. (We say that h: B→A extends f : X→A when therestriction of h to X is f.)

As a convenient shorthand a free C-algebra on no generators can also becalled an initial C-algebra. An initial C-algebra has exactly onehomomorphism to every C-algebra.

Before proceeding to the examples it is worthwhile pointing out animportant basic property of free algebras as defined from the semanticside.

By way of proof, pick any bijection f : X→Y. This, its inverse fʹ′: Y→X, and

Two free algebras B, Bʹ′ on respective generator sets X, Y havingthe same cardinality are isomorphic.

Vaughan Pratt


the two identity functions on respectively X and Y, form a system of fourfunctions closed under composition. Each of these functions is from agenerator set to an algebra and therefore has a unique extension to ahomomorphism. These four homomorphisms are also closed undercomposition. The one from B to itself extends the identity function on Xand therefore must be the identity homomorphism on B (since the latterexists and its restriction to X is the identity function on X). Likewise thehomomorphism from G to G is an identity function. Hence thehomomorphisms between B and G compose in either order to identities,which makes them isomorphisms. But this is what it means for B and Bʹ′ tobe isomorphic.

This fact allows us to say the free algebra on a given set, thinking ofisomorphic algebras as being "morally" the same. Were this not the case,our quotient construction would be incomplete as it produces a unique freealgebra, whereas the above definition of free algebra allows any algebraisomorphic to that produced by the quotient construction to be consideredfree. Since all free algebras on X are isomorphic, the quotient constructionis as good as any, and is furthermore one way of proving that they exist. Italso establishes that the choice of set of variables is irrelevant except forits cardinality, as intuition would suggest.

6.1 Free monoids and groups

Take C to be the class of monoids. The term algebra determined by thebinary operation symbol and the constant symbol for identity can beviewed as binary trees with variables and copies of the constant symbol atthe leaves. Identifying trees according to associativity has the effect offlattening the trees into words that ignore the order in which the operationwas applied (without however reversing the order of any arguments). Thisproduces words over the alphabet X together with the identity. The identitylaws then erase the identities, except in the case of a word consisting only

Algebra


of the identity symbol, which we take to be the empty word.

Thus the monoid of finite words over an alphabet X is the free monoid onX.

Another representation of the free monoid on n generators is as an infinitetree, every vertex of which has n descendants, one for each letter of thealphabet, with each edge labeled by the corresponding letter. Each vertex vrepresents the word consisting of the letters encountered along the pathfrom the root to v. The concatenation of u and v is the vertex arrived at bytaking the subtree whose root is the vertex u, noticing that this tree isisomorphic to the full tree, and locating v in this subtree as though it werethe full tree.

If we ignore the direction and labels of the edges in this tree we can stillidentify the root: it is the only vertex with n edges incident on it, all othervertices have n+1, namely the one incoming edge and the n outgoing ones.

The free commutative monoid on a set is that monoid whose generatorsbehave like letters just as for free monoids (in particular they are stillatoms), but which satisfy the additional law ugv = vu. We make furtheridentifications, e.g. of "dog" and "dgo". Order of letters in a word is nowimmaterial, all that matters is how many copies there are of each letter.This information can be represented as an n-tuple of natural numberswhere n is the size of the alphabet. Thus the free commutative monoid onn generators is Nn, the algebra of n-tuples of natural numbers underaddition.

It can also be obtained from the tree representation of the free monoid byidentifying vertices. Consider the case n = 2 of two letters. Since theidentifications do not change word length, all identifications are ofvertices at the same depth from the root. We perform all identificationssimultaneously as follows. At every vertex v, identify v01 and v10 and their

Vaughan Pratt


subtrees. Whereas before there were 2n vertices at depth n, now there aren+1. Furthermore instead of a tree we have the upper right quadrant of theplane, that is, N², rotated 135 degrees clockwise, with every vertex v at thetop of a diamond whose other vertices are v0 and v1 at the next leveldown, and the identified pair v01 = v10 below both.

To form the free group on n generators, first form the free monoid on 2ngenerators, with generators organized into complementary pairs each theinverse of the other, and then delete all adjacent complementary pairs fromall words.

This view is not particularly insightful. The group counterpart of the treerepresentation does a better job of presenting a free group. Consider thefree group on n = 2 generators A and B. We start with the free monoid on 4generators A, B, a, b where a is the inverse of A and b that of B. Everyvertex of this tree has 4 descendants. So the root has degree 4 and theremaining vertices have degree 5: every vertex except the root has oneedge going in, say the generator a, and four out. Consider any nonrootvertex v. The effect of deleting adjacent complementary pairs is to identifythe immediate ancestor of v with one of the four descendants of v, namelythe one that makes the path from the ancestor to the descendant acomplementary pair. For every nonroot vertex v these identificationsreduce the degree of v from 5 to 4. The root remains at degree 4.

So now we have an infinite graph every vertex of which has degree 4.Unlike the tree for the free monoid on 2 generators, where the root istopologically different from the other vertices, the tree for the free groupon 2 generators is entirely homogenous. Thus if we throw away the vertexlabels and rely only on the edge labels to navigate, the graph is perfectlyhomogeneous.

This homogeneity remains the case for the free abelian group on 2

Algebra


generators, whose vertices are still of degree 4. However the additionalidentifications turns it from a tree (a graph with no cycles) to a grid whosevertices are the lattice points of the plane. That is, the free group on 2generators is Z², and on n generators Zn. The edges are the line segmentsjoining adjacent lattice points.

6.2 Free rings

With no generators the free monoid, free group, and free ring are all theone-element algebra consisting of just the additive identity 0. A ring withidentity means having a multiplicative identity, that is, a word ε. But thismakes ε a generator for the additive group of the ring, and the free abeliangroup on one generator is the integers. So the free ring with identity on nogenerators is the integers under addition and now multiplication.

The free ring on one generator x must include x², x³, etc. by multiplication,but these can be added and subtracted resulting in polynomials such as 7x³−3x²+2x but without a constant term, with the exception of 0 itself. Thedistributivity law for rings means that a term such as (7x+x²)(2x³+x) can beexpanded as 7x²+x³+14x4+2x 5. It should now be clear that these are justordinary polynomials with no constant term; in particular we are missingthe zero-degree polynomial 1 and so this ring has no multiplicativeidentity. However it is a commutative ring even though we did not specifythis. The free ring with identity on one generator introduces 1 as themultiplicative identity and becomes the ordinary one-variable polynomialssince now we can form all the integers. Just as with monoids, the free ringwith identity on two generators is not commutative, the polynomials xgyand ygx being distinct. The free commutative ring with identity on twogenerators however consists of the ordinary two-variable polynomials overthe integers.

From the examples so far one might conclude that all free algebras on one

Vaughan Pratt


or more generators are infinite. This is by no means always the case; ascounterexamples we may point to a number of classes: sets, pointed sets,bipointed sets, graphs, undirected graphs, Boolean algebras, distributivelattices, etc. Each of these forms a locally finite variety as defined earlier.

6.3 Free combinatorial structures

A pointed set is an algebra with one constant, say c. The free pointed seton x and y has three elements, x, y, and c. A bipointed set is an algebrawith two constants c and d, and the free bipointed set on x and y then hasfour elements, x, y, c, and d.

Graphs, of the oriented kind arising in say automata theory where multipleedges may connect the same two vertices, can be organized as algebrashaving two unary operations s and t satisfying s(s(x)) = t(s(x)) = s(x) andt(t(x)) = s(t(x)) = t(x). The free graph on one generator x has threeelements, x, s(x), and t(x), constituting respectively an edge and its twoendpoints or vertices. In this framework the vertices are the elementssatisfying s(x) = x (and hence t(x) = x since x = s(x) = t(s(x)) = t(x)); allother elements constitute edges. The free graph on n generators consists ofn such edges, all independent. Other graphs arise by identifying elements.There is no point identifying an edge with either another edge or a vertexsince that simply absorbs the first edge into the second entity. This leavesonly vertices; identifying two vertices yields a single vertex common totwo edges, or to the same edge in the identification s(x) = t(x) creating aself-loop.

The term "oriented" is to be preferred to "directed" because a directedgraph as understood in combinatorics is an oriented graph with theadditional property that if s(x) = s(y) and t(x) = t(y) then x = y; that is, onlyone edge is permitted between two vertices in a given direction.

Unoriented graphs are defined as for graphs with an additional unary

Algebra


operation g satisfying g(g(x)) = x and s(g(x)) = t(x) (whence s(x) =s(g(g(x))) = t(g(x))). The free undirected graph on x consists of x, s(x), t(x),and g(x), with the pair x, g(x) constituting the two one-way lanes of a two-lane highway between s(x) = t(g(x)) and and t(x) = s(g(x)). Identificationof elements of undirected graphs works as for their oriented counterparts:it is only worth identifying vertices. However there is one interesting twisthere: vertices can be of two kinds, those satisfying x = g(x) and those not.The latter kind of vertex is now asymmetric: one direction of thebidirectional edge is identified with its vertices while the other one formsan oriented loop in the sense that its other direction is a vertex. Thisphenomenon does not arise for undirected graphs defined as thosesatisfying "if s(x) = s(y) and t(x) = t(y) then x = y."

6.4 Free logical structures

Boolean algebras are traditionally defined axiomatically as complementeddistributive lattices, which has the benefit of showing that they form avariety, and furthermore a finitely axiomatizable one. However Booleanalgebras are so fundamental in their own right that, rather than go to thetrouble of defining lattice, distributive, and complemented just for thispurpose, it is easier as well as more insightful to obtain them from theinitial Boolean algebra. It suffices to define this as the two-element set {0,1}, the constants (zeroary operations) 0 and 1, and the 2²² = 16 binaryoperations. A Boolean algebra is then any algebra with those 16 operationsand two constants satisfying the equations satisfied by the initial Booleanalgebra.

An almost-definitive property of the class of Boolean algebras is that theirpolynomials in the initial Boolean algebra are all the operations on thatalgebra. The catch is that the inconsistent class consisting of only the one-element or inconsistent algebra also has this property. This class is easilyruled out however by adding that Boolean algebra is consistent. But just

Vaughan Pratt


barely — adding any new equation to Boolean algebra (withoutintroducing new operations) axiomatizes the inconsistent algebra.

Sheffer has shown that the constants and the 16 operations can begenerated as polynomials in just one constant, which can be 0 or 1, andone binary operation, which can be NAND, ¬(x∧y), or NOR, ¬(x∨y). Anysuch sufficient set is called a basis. Along the same lines Stone has shownthat conjunction, exclusive-or, and the constant 1 form a basis. Thesignificance of Stone's basis over Sheffer's is that Boolean algebrasorganized with those operations satisfy all the axioms for a commutativering with identity with conjunction as multiplication and exclusive-or asaddition, as well as the law x² = 1. Any ring satisfying this last condition iscalled a Boolean ring. Boolean rings are equivalent to Boolean algebras inthe sense that they have the same polynomials.

An atom of a Boolean algebra is an element x such that for all y, x∧y iseither x or 0. An atomless Boolean algebra is one with no atoms.

There is exactly one Boolean algebra of cardinality every finite power of2, and it is isomorphic to the Boolean algebra of a power set 2X of thatcardinality under the set operations of union, intersection, and complementrelative to X. Hence all finite Boolean algebras have cardinality a power of2. This situation changes with infinite Boolean algebras; in particularcountable Boolean algebras exist. One such is the free Boolean algebra oncountably many generators, which is the only countable atomless Booleanalgebra. The finite and cofinite (complement of a finite set) subsets of theset N of natural numbers form a subalgebra of the powerset Booleanalgebra 2N not isomorphic to the free Boolean algebra, but it has atoms,namely the singleton sets.

The free Boolean algebra F(n) on n generators consists of all 2²n n-aryoperations on the two-element Boolean algebra. Boolean algebras

Algebra


therefore form a locally finite variety.

The equational theory of distributive lattices is obtained from that ofBoolean algebras by selecting as its operations just the monotone binaryoperations on the two-element algebra, omitting the constants. These arethe operations with the property that if either argument is changed from 0to 1, the result does not change from 1 to 0. A distributive lattice is anymodel of those Boolean equations between terms built solely withmonotone binary operations. Hence every Boolean algebra is a distributivelattice.

Distributive lattices can be arbitrarily "thin." At the extreme, any chain(linear or total order, e.g. the reals standardly ordered) under the usualoperations of max and min forms a distributive lattice. Since we haveomitted the constants this includes the empty lattice, which we have notexcluded here as an algebra. (Some authors disallow the empty set as analgebra but this proscription spoils many good theorems without gainingany useful ones.) Hence there exist distributive lattices of every possiblecardinality.

Every finite-dimensional vector space is free, being generated by anychoice of basis. This extends to infinite-dimensional vector spacesprovided we accept the Axiom of Choice. Vector spaces over a finite fieldtherefore form a locally finite variety when scalar multiplication isorganized as one unary operation for each field element.

6.5 Free algebras categorially

We now consider how free algebras are organized from the perspective ofcategory theory. We defined the free algebra B generated by a subset X ofB as having the property that for every algebra A and every valuation f :X→A, there exists a unique homomorphism h: B→A. Now everyhomomorphism h: B→A necessarily arises in this way, since its restriction

Vaughan Pratt


to X, as a function from X to A, is a valuation. Furthermore every functionf : X→A arises as the restriction to X of its extension to a homomorphism.Hence we have a bijection between the functions from X to A and thehomomorphisms from B to A.

Now the typing here is a little casual, so let us clean it up. Since X is a setwhile A is an algebra, f is better typed as f : X→U(A) where U(A) denotesthe underlying set of A. And the relationship of X to B is better understoodwith the notation B = F(X) denoting the free algebra generated by the setX. So U maps algebras to sets while F maps sets to algebras. F and U arenot in general inverses of each other, but they are nonetheless related in away we now make precise.

The notation C(A, B) is generally used to denote the set of allhomomorphisms from A to B. And the set of all functions from set X to setY can be understood as the particular case Set(X, Y) of this conventionwhere C is taken to be the class Set of all sets, which we can think of asdiscrete algebras, that is, algebras with no structure. A class of algebrasalong with a specified set of homomorphisms between any two of itsmembers is an instance of a category. The members of the class are calledthe objects of the category while the homomorphisms are called themorphisms.

The bijection we have just observed can now be stated as

Such a bijection is called an adjunction between Set and C. f : Set→C andU: C→Set are respectively the left and right adjoints of this adjunction;we say that F is left adjoint to (or of) U and U right adjoint to F.

We have only described how F maps sets to algebras, and U mapsalgebras to sets. However F also maps functions to homomorphisms,

C(F(X), A) ≅ Set(X, U(A))

Algebra


mapping f to its unique extension as a homomorphism, while U mapshomomorphisms to functions, namely the homomorphism itself as afunction. Such maps between categories are instances of functors.

In general a category C consists of objects a, b, c and morphisms f : a→b,together with an associative composition law for "composable" morphismsf : b→c, g: a→b yielding the morphism fgg: a→c. Furthermore everyobject a has an identity element 1a: a→a which whenever composablewith a morphism f (on one side or the other) composes with it to yield f. Afunctor F : C→D maps objects of C to objects of D and morphisms of C tomorphisms of D, such that F(fgg) = F(f)F(g) and F(1a) = 1F(a). That is,functors are "homomorphisms of categories," preserving composition andidentities.

With no further qualification such a category is considered an abstractcategory. The categories we have been working with are concrete in thesense that they come with a given underlying set or forgetful functor U:C→Set. That is, algebras are based on sets, homomorphisms are certainfunctions between these sets, and U simply "forgets" the algebraicstructure. Such forgetful functors are faithful in the sense that for any twomorphisms f, g: a→b of C, if U(f) = U(g) then f = g, i.e. U does notidentify distinct homomorphisms. In general a concrete category is definedas a category C together with a faithful forgetful functor U: C→Set.

Categories themselves admit a further generalization to 2-categories asalgebras over two-dimensional graphs, with associative composition of 1-cells generalized to the 2-associative pasting of 2-cells. A furthersimplification of the free-algebra machinery then obtains, namely viaabstract adjunctions as the natural 2-dimensional counterpart ofisomorphisms in a category, which in turn is the natural 1-dimensionalcounterpart of equality of elements in a set, the 0-dimensional idea thattwo points can turn out to be one. This leads to a notion of abstract monad

Vaughan Pratt


as simply the composition of an adjoint pair of 1-cells, one of which is the1-cell abstracting the functor F that manufactures the free algebra F(V)from V. Ordinary or concrete monads arise as the composition of functorsas concrete 1-cells of a 2-category of categories.

Bibliography

Harold R. Jacobs, Elementary Algebra, 876pp, W.H. Freeman, 1979.I.N. Herstein, Topics in Algebra, 2nd ed, 400pp, Wiley, 1975.

Other Internet Resources

Many algebraic topics, at Mathworld, Wolfram Research.Linear algebra, at Answers.com

Related Entries

Boolean algebra: the mathematics of | category theory

Copyright © 2014 by the author Vaughan Pratt

Algebra


Documents

Algebra Sc