Linear Types and ApproximationMichael Huth Achim Jung Klaus KeimelJanuary 11, 1999AbstractWe study continuous lattices with maps which preserve all supremarather than only directed ones. We introduce the subclass of FS-latticeswhich turns out to be �-autonomous, and in fact maximal with this prop-erty. FS-lattices are studied in the presence of distributivity and algebraic-ity. The theory is extremely rich with numerous connections to classicalDomain Theory, complete distributivity, Topology, and models of LinearLogic.1 IntroductionThe work reported in this paper derives its motivation from at least three di�er-ent directions. Firstly, there is the theory of autonomous (or symmetric monoidalclosed) categories as described extensively in [EK66]. These are abstractions ofthe frequent phenomenon in algebra of the set of homomorphisms between twostructures being a structure of the same kind again without the internal homfunctor interacting with the product in the usual way. The correspondence as itis expressed in Linear Algebra, then, is between bilinear maps and tensor productsrather than between linear maps and products. In [Bar79], the abstract theoryof symmetric monoidal closed categories is extended with a duality derived froma dualizing object ?. Again, algebra provides a number of motivating exam-ples. One of these is the category SUP of complete lattices and sup-preservingfunctions.1 In the present paper we augment the objects of this category witha notion of \approximation" in the sense of Domain Theory [AJ94]. It will beshown that the full subcategory CL of continuous lattices is not closed and oneof our main results characterizes the largest closed full subcategory of CL (underone extra condition). The result is reminiscent of similar theorems for cartesianclosed categories [Smy83, Jun90]; it would be very interesting to �nd a deeperreason for this similarity.1In fact, Barr works with in�ma rather than suprema but this di�erence is immaterial.1

From a di�erent perspective, this paper introduces a new model for ClassicalLinear Logic [Gir87]. One the surface of it, this construction seems fairly straight-forward, given the general theory of �-autonomous categories as explicated in[Bar91]. We choose the modality ! to be that of all Scott-closed subsets of thelattice with the goal in mind to get Scott-continuous maps in the correspondingco-Kleisli category. Rather pleasingly, the dual modality ? has a meaningful in-terpretation in its own right rather than just being the de Morgan dual of !; ityields precisely the so-called Smyth-powerdomain [Smy78]. One may see this asa vindication of the move to approximated lattices, as such a characterization isnot available in the bigger category SUP. ([AJ94] contains other instances ofthis phenomenon.)Finally, one may see this paper as an attempt to achieve a linear decomposi-tion of Scott-continuous functions along the lines of Girard's original constructionof coherence spaces and stable maps. It is then interesting to see that certainconcepts of Domain Theory still apply, certifying to their robustness and gener-ality.The structure of the paper is as follows. We recall the algebraic traditionwhich led to the theory of �-autonomous categories in Section 2. In Section 3 wegive some details of Barr's example SUP for a �-autonomous category consistingof complete lattices and suprema preserving functions. It is the ambient categoryfor the remainder of the paper. Section 4 introduces the main objects of study,linear FS-lattices. They are de�ned in analogy to FS-domains, [Jun90], and, asin the Scott-continuous setting, they provide a closed category of approximatedobjects. In fact, we are able to show that they are a maximal choice when acertain further condition (called \leanness") is assumed. FS-domains are subse-quently augmented with two (independent) properties: distributivity (Section 5)and algebraicity (Section 7). In both cases, we obtain additional information: dis-tributive FS-domains turn out to be completely distributive and they form notonly a �-autonomous but a compact closed category. Algebraic FS-domains areshown to be exactly the bi�nite ones (in the linear sense), and a fairly involvedargument in Subsection 7.3 shows that algebraic FS-domains are the maximal�-autonomous full subcategory of SUP whose objects are algebraic. A numberof parallels between the Scott-continuous and the linear setting are pointed outin the remainder of Section 7.In between, in Section 6, we show how to build a Benton-model of LinearLogic with the ingredients of Domain Theory. The development is extremelysmooth and we would like to claim that the model is a natural yet non-trivialone. We were particularly pleased to �nd the connection between modalities andpowerdomains mentioned before. Although Section 6 refers to distributivity atsome point, it can be read directly after Section 4.Section 8 indicates how the theory could be extended from lattices to Scott-domains. For the sake of brevity, we have refrained from a detailed exposition.Section 9 refers to further interesting discoveries about FS-domains, which were2

made more recently.In our notation for domain theoretic concepts we follow [AJ94]; relevantbackground information on continuous lattices can be found there as well asin [GHK+80].2 Categorical preliminariesIf K is a class of algebraic structures and A;B;C are objects in K, the one callsa map �:A�B ! C a bihomomorphism if for every a 2 A, b 2 B the functions�(a; ): y 7! �(a; y); and�( ; b): x 7! �(x; b)are homomorphisms of K. The prime example is vector spaces and bilinear maps.A category is an abstract version of \class of structures of the same kind andtheir homomorphisms". However, the de�nition of a bihomomorphism seemsto require an explicit reference to elements. Also, the map � itself is certainlyexternal to the category at hand.A slight rede�nition of bihomomorphism is more amenable to a categoricaltreatment. Instead of �:A � B ! C, we consider �0:A ! (B ! C) given by�0(x)(y) := �(x; y). If we assume that the set (B ! C) of homomorphisms isitself a structure of the same kind as A;B and C, through a pointwise de�nitionof the operations, then bihomomorphisms �:A � B ! C and homomorphism�0:A ! (B ! C) are in one-to-one correspondence. These two conditions areindeed satis�ed for vector spaces and also for the objects under considerationhere, complete lattices with sup-preserving maps.Categorically, one requires an object > and an internal hom-functor ( ! ),contravariant in the �rst and covariant in the second argument, to model therequirement that the set of homomorphisms quali�es as a structure. In order torecognize the object (A ! B) as the set of homomorphisms from A to B onerequires certain natural transformations and equivalences, to wit(> ! A) �= A> ��! (A! A)(B ! C) ��! ((A! B)! (A! C))subject to a number of axioms [EK66]. A category with these properties is calledclosed. In a closed category we may replace \bihomomorphism" with \morphismfrom A to (B ! C)". See [BN76] for an in-depth discussion.A closed category is called symmetric closed if the (A ! (B ! C)) and(B ! (A! C)) are naturally isomorphic.From Linear Algebra we know that bilinear maps A� B ! C are in one-to-one correspondence with linear maps AB ! C, where denotes the tensor3

product of vector spaces. Abstractly, then, the presence of a \tensor product"gives us an alternative way of coding bihomomorphisms. To make this precise,one stipulates that be a bifunctor for which B is left adjoint to (B ! ),or, equivalently, (A B ! C) and (A ! (B ! C)) are naturally isomorphic.In addition to this, the abstract tensor product is required to be associative andto have a unit I subject to a number of coherence axioms [EK66, Mac71]. Withthis additional data, we arrive at a monoidal closed category. In a monoidalclosed category, which is also symmetric in the sense above, the tensor productis commutative, AB �= B A. Together, one speaks of a symmetric monoidalclosed or autonomous category.One last remark: Not every algebraic theory allows us to internalize the hom-functor (non-Abelian groups are an example) and even if it does, a suitable tensorproduct may not exist. Beyond these two obstacles, a further one needs to beovercome for a category to be cartesian closed, namely, it must be the case thatbihomomorphisms are already homomorphisms. The category SET quali�es fortrivial reasons; in the case ofDCPO (directed-complete partial orders and Scott-continuous functions) this is one of the fundamental lemmas of its theory [AJ94,3.2.6].In [Bar79], Michael Barr studies the situation where an autonomous categoryis equipped with an internal duality, that is, where there exists an object ? suchthat A and ((A! ?)! ?) are naturally isomorphic for all objects A. WritingA? for (A! ?), one gets the following equivalences:(A! B) �= (B? ! A?) (1)A B �= (A! B?)? (2)without making any further assumptions. A category with these properties,dubbed �-autonomous by Barr, provides a model for the multiplicative part ofLinear Logic, [Gir87, Bar91].3 SUP as a model of Linear LogicThe category SUP of complete lattices and suprema preserving maps was men-tioned as an example for a �-autonomous category in [Bar79]. For our purposesbelow, it will be necessary to have some understanding of the concrete structureof the various connectives in SUP. We will also have to adjust the categoricalnotation to this particular setting.De�nition 1 Let A and B be complete lattices and f a map from A to B. Wecall f linear if it preserves all suprema, f(WX) = W f(X), X � A. We writef :A �� B in this situation. The set of all linear maps between A and B, orderedpointwise, is denoted by (A �� B). 4

Complete associativity of the supremum operation in lattices entails that thefunction space (A �� B) is again a complete lattice.Every linear map f :A �� B has an upper adjoint f �:B ! A [AJ94, Sect. 3.1.3],[GHK+80, Chapter IV]. It is given byf �(y) :=_ fx j f(x) � yg :Alternatively, the correspondence between f and f � may be encoded in the equiv-alence f(x) � y () x � f �(y) : (3)From this we glean that the assignment f 7! f � is order reversing2. Hence, if weview f � as a map fromBop to Aop, we get a linear function and the correspondencef 7! f �: (A �� B)! (Bop �� Aop) is in fact an order isomorphism.There is only one possibility for a dualizing object in SUP, and this is thetwo-element lattice 2. For the dual A? of a complete lattice A with respect to? = 2, we haveA? = (A �� 2) �= (2op �� Aop) �= (2 �� Aop) �= Aop ;where the last isomorphism holds because the bottom element of 2 must bemapped onto the bottom element of Aop by any linear function and the topelement can be mapped onto any element of Aop whatsoever.From now on, we will write Aop instead of A? and 2 instead of ? to avoidconfusion with the established notation for the least element of a domain. Also,we will use the symbols, �, _, etc, as they apply to A even when we speak of Aop.For the tensor product we take Equivalence (2) as the (necessary) de�nition:AB := (A �� Bop)op. Concretely, a linear map r from A to Bop corresponds toan antitone map from A to B which translates suprema into in�ma. The upperadjoint r�:Bop ! A, if viewed as a function from B to A, has exactly the sameproperty. Together, (r; r�) form a Galois-connection between A and B. Any pairof maps between complete lattices satisfyingr(x) � y () x � s(y) ; x 2 A; y 2 B (4)is of this kind.The de Morgan dual of , denoted O (\par") is given by the set of linearfunctions from Aop to B. Maps r:Aop �� B together with their adjoints r�:Bop ��A form pairs (r; r�) which are completely characterized by the equivalencer(x) � y () x � s(y) ; x 2 A; y 2 B :2Assume f � g. From g�(y) � g�(y) get g(g�(y)) � y and hence f(g�(y)) � y. Thereforeg�(y) � f�(y). 5

As noted in [Bar79], AOB can be di�erent from AB, even for �nite lattices Aand B. In fact, it is distributivity, not �niteness, which renders O and equal,as we will see in Section 5.It is quite enjoyable to explore what the abstract equivalences of a �-autonomouscategory amount to in the case of SUP. For example, the symmetry of the ten-sor product is e�ected by switching to the other half of a Galois-connection. Thenatural isomorphism between (A B �� C) and (A �� (B �� C)) is encoded inthe equation ��(c)(a) = (a)�(c)� 2 (A B �� C) 2 (A �� (B �� C))in which one side completely determines the other.Besides the multiplicatives of Linear Logic, which are all faithfully modelledbecause SUP is �-autonomous, we can also study the additives N and �. InSUP, these are both modelled by cartesian product (which is also the coproductbecause (A � B)op �= Aop � Bop), with the one-element lattice representing theunits.Since the interpretations of N and � coincide, our model satis�es all distribu-tivity laws of the form Am(B aC), where m 2 f;Og, a 2 fN;�g, i.e. it is fullydistributive. This property was noted in [Hut95b] already. It has recently beenstudied from a proof-theoretic point of view in [Len98].4 Adding approximationWe come to the main objective of this paper, which is to enrich the objects ofBarr's category SUP with a domain theoretic notion of approximation; that is, toconsider continuous lattices. We are faced with an immediate di�culty, becausethe category CL of continuous lattices and linear maps is not closed.Example 2 LetM1 be the lattice of the discretely ordered set of natural numbersextended with a least and a largest element (see Figure 1 below). In the linearfunction space (M1 �� M1) we look at the identity id. Because all maps ofthis space are sup-preserving, there is only one function below id, namely, theconstant bottom function. If (M1 �� M1) were continuous, then id would haveto be a compact element. However, we have the following chain of maps whosesupremum exceeds id without any of its elements being above id:fn:M1 !M1; n 2 Nfn(?) = ?; fn(>) = >fn(m) = � >; if m � n;m + 1; otherwise.6

?>0 1 2 3

Figure 1: The lattice M1.A similar problem arises in Domain Theory. There one has the cartesianclosed category DCPO whose full subcategories of continuous, respectively alge-braic, domains are not closed. By restricting these categories further one recoversclosedness. Examples are Scott-domains, SFP-domains, etc., see [AJ94, Chap-ter 4] for more details. In the same vein, we will now exhibit a full subcategoryof CL which is closed.De�nition 3 ([Jun90]) A function f :A ! A on a partially ordered set A issaid to be �nitely separated from idA, if there exists a �nite subset M of A suchthat for all x 2 A there exists m 2M with f(x) � m � x.For a complete lattice A to be an FS-lattice we require the existence of adirected family D of linear functions on A whose supremum equals idA. Let FSdenote the full subcategory of SUP whose objects are FS-lattices.Obviously, every �nite lattice is in FS because we can choose D = fidg inthis case. As for in�nite examples, we will see in Section 5 below that everycompletely distributive lattice is in FS. At this point, however, it is necessary tojustify our de�nition by showing that FS-lattices are indeed continuous. We let[A! B] denote the complete lattice of all Scott-continuous functions f :A! Bin the pointwise order. Note that (A �� B) is a subset of [A! B] closed underall suprema.Lemma 4 Let A be a complete lattice. If a Scott-continuous function f 2[A! A] is �nitely separated from idA, then f(x)� x for all x 2 A.Proof. LetM be the �nite subset of A which separates f from idA. Given x 2 Aand a directed set D � A with x � W "D let Dm := fd 2 D j f(d) � m � dg,m 2M . By assumption we have D = Sm2M Dm and so at least one Dm0 must beco�nal in D. Hence we get f(x) � f(W "D) = f(W"Dm0) = W "f(Dmo) � m0 � dfor any d 2 Dm0 .Corollary 5 FS-lattices are continuous.Let us now show that FS carries enough structure to model all of Linear Logic.As we know from Section 3, the whole structure of a �-autonomous category isderived from the function space. The following is therefore crucial.Lemma 6 Let A and B be FS-lattices. Then (A �� B) is also an FS-lattice.7

Proof. Let D � (A �� A) and E � (B �� B) be directed sets with W "D = idAand W"E = idB such that all f 2 D and g 2 E are �nitely separated fromthe respective identities. For f 2 D, g 2 E and Mf , Mg the respective �niteseparating sets, we will show that �2f;g, where �f;g(h) = g � h � f , is �nitelyseparated from id(A��B). This su�ces to prove the result because W "�2f;g is equalto id(A��B). So let f 2 D, g 2 E be given. We de�ne an equivalence relation �on (A �� B) byh1 � h2 :, 8m 2Mf : "g(h1(m)) \Mg = "g(h2(m)) \Mg:As Mf and Mg are �nite, there are only �nitely many equivalence classes on(A �� B). Let K be a set of representatives of these classes. We claim that the�nite set �f;g(K) separates �2f;g from id(A��B). Given h 2 (A �� B), let kh be thecorresponding representative in K. For a 2 A, we computeh(a) � h(mf ) for some mf 2Mf with f(a) � mf � a� mg for some mg 2Mg with g(h(mf)) � mg � h(mf )� g(kh(mf)) as g(h(mf)) � mg and h � kh� g(kh(f(a))) as f(x) � mf :By symmetry, we obtain kh � �f;g(h), so h � �f;g(kh) � �2f;g(h).A similar proof, for FS-domains, appeared �rst in [Jun90].Theorem 7 FS is a �-autonomous full subcategory of SUP. Furthermore, it isclosed under cartesian products.Remember that the order dual of a lattice, Aop, can be expressed as a linearfunction space: Aop �= (A �� 2), so the preceding theorem says in particular thatwith A we automatically have that Aop is FS again.Let us now attempt to show that FS is indeed the largest full subcategoryof continuous lattices of SUP which is closed. Finiteness, which is part of thede�nition of an FS-lattice, will have to come from a compactness argument. Inother words, we will have to work with topological concepts as well as order the-oretic ones. The topology which is appropriate for our purposes is the patch-or Lawson-topology, because it is compact Hausdor� on a continuous lattice,[GHK+80, Theorem III-1.10]. It is a re�nement of the Scott-topology and gener-ated by Scott-open subsets and complements of Scott-compact upper subsets.Now, for a complete lattice A it is easy to see that every Scott-compact upperset C � A is closed with respect to the Scott-topology �A on Aop because adownward directed set (xi)i2I gives rise to a directed collection (A n #xi)i2I ofScott-open sets, resulting in a compactness argument if the in�mum of (xi)i2Iis assumed not to be in C. The converse is not necessarily true: Consider thelatticeM1 from Example 2; every upper set inM1 is closed with respect to �Mop1but only �nite upper sets are compact with respect to �M1 . Let us say that acomplete lattice A is lean if every �Aop-closed subset is �A-compact.Somewhat surprisingly, leanness is a self-dual concept in our setting:8

Lemma 8 Let A be a bicontinuous lattice. Then A is lean if and only if Aop islean.Proof. Let us denote the join of the two Scott-topologies by �2. It is a re�ne-ment of both Lawson-topologies �A and �Aop. Under the assumption of continu-ity, the Lawson-topology is compact Hausdor�. In this setting, for A to be leanmean nothing else but �A = �2. So assuming A to be lean renders �2 a compactHausdor� re�nement of the compact Hausdor� topology �Aop . It is a standardtopological result that the two topologies must coincide in this case.Remark 9 The previous lemma holds already if A and Aop are assumed to besober spaces in their Scott-topologies, because then the so-called patch topologiesare then compact Hausdor�. We will, however, not need this generality.Lemma 10 FS-lattices are lean.Proof. Let C be a �Aop-closed subset of the FS-lattice A and let (fi)i2I be anapproximating family of �nitely separated linear maps. For each i 2 I let Mibe the �nite separating set. We have that C is contained in "Ni where Ni =fm 2Mi j 9x 2 C:fi(x) � m � xg. Each "Ni is �A-compact as it is generated bya �nite set. The intersection C 0 of all "Ni, i 2 I, contains C and is �A-compactagain because A is a complete lattice, [AJ94, Theorem 4.2.18]. All we need toshow is that C 0 = C.To this end let a be in the �Aop-open set A n C. Since the family of upperadjoints (f �i )i2I is approximating from above there exists i0 2 I such that f �i0(a) 2A n C. The corresponding fi0 maps C into A n #a because fi0(x) � a impliesx � f �i0(a). It follows that "Ni0 does not contain a.After these preliminaries, let us now press on towards the promised maximal-ity result.Lemma 11 Let A be a complete lattice and f � g in (A �� A). Then f(a) �g(a) for all a 2 A.Proof. Let g(a) � W "i2I xi be given. De�nefi(x) := 8<: ?A; x = ?A;xi; x � a;>A; otherwise.Then (fi)i2I is directed in (A �� A) and g � W"i2I fi. Since f � g in (A �� A)we have f � fj for some j 2 I and f(a) � fj(a) = xj as desired.Corollary 12 Let A be a complete lattice such that (A �� A) is continuous.Then both A and Aop are continuous. 9

Proof. For A this follows directly from the previous lemma. It is true for Aopas well because (A �� A) and (Aop �� Aop) are isomorphic.Lemma 13 Let A be a lean continuous lattice with continuous linear functionspace (A �� A). If f is way-below idA in (A �� A), then f is �nitely separatedfrom idA.Proof. The continuity of (A �� A) and the Scott-continuity of compositionimply the existence of some g � idA with f � g � g. As h 7! h�: (A ��A) ! (Aop �� Aop) is an order isomorphism, we obtain g� � idAop in (Aop ��Aop). By the previous lemma, g�(a) � a in Aop for all a 2 A. Thus, Oa :=fb 2 Aop j g�(a)� b in Aopg contains a and is Scott-open in Aop. Since A is lean,this set is also �A-open. The continuity ofA ensures that Ua := fe 2 A j g(a)� e in Agis Scott-open in A; again, it contains a. Thus, Va := Oa \ Ua is a �A-open setcontaining a.The topology �A is compact as A is continuous. Therefore, the open coverSa2A Va of A has a �nite subcover A = Sm2M Vm. For a 2 A, we have a 2 Vmfor some m 2 M . In particular, this guarantees the inequalities g(m) � a anda � g�(m). The latter is equivalent to g(a) � m, so f(a) � g(g(a)) � g(m) � ashows that g(M) is a �nite set separating f from idA.As a direct consequence of this lemma we get our �rst main result.Theorem 14 FS is the largest (full) �-autonomous subcategory of SUP whoseobjects are lean and continuous.It is slightly unsatisfactory that we need to refer to leanness in the statementof this theorem. Indeed, in Section 7.3 we dispense with this condition in thespecial case of algebraic lattices. The proof, as we will see, is rather technical andmakes vital use of the abundance of compact elements. It would be desirable tohave a more conceptual account of this result which | one hopes | would thenalso apply to continuous lattices. We leave this as an open problem.5 DistributivityThe aim of this section is to study the subcategory CD of SUP whose objectsare completely distributive lattices. Before we do so, we need to record somefundamental properties of these lattices.It was discovered very early on in the history of continuous lattices that thereis a strong connection between the notions of approximation and distributivity,[Sco72] and [GHK+80, Theorem I-2.3]. In the case of completely distributivelattices this connection was noted even earlier in the work of G.N. Raney, [Ran53].Let us review the main points. 10

De�nition 15 Let x; y be elements of a complete lattice A. We say that a0 iscompletely below a (and write a0n a) if for every subset X of A we have thata � WX implies a0 � x for some x 2 X.This, of course, is the same as the de�nition of the way-below relation witharbitrary subsets replacing the directed ones. The elementary properties ofnare the same as for � and their proofs are completely analogous (and simpler):Proposition 16 For any complete lattice A and a; a0; b; b0 2 A the following aretrue:1. a0n a implies a0 � a;2. a0 � an b � b0 implies a0n b0;3. ?n a if and only if ? 6= a.We can now de�ne a complete latticeA to be super-continuous if every elementof A is the supremum of elements completely below it. However, super-continuityis equivalent to complete distributivity:Theorem 17 (Raney) A complete lattice A is completely distributive if andonly if for all a 2 A, a = W fa0 2 A j a0n ag holds.Corollary 18 1. A complete lattice A is super-continuous if and only if Aopis super-continuous.2. Completely distributive lattices are bicontinuous.The corollary says that we get approximation from both sides automati-cally in super-continuous lattices. Observe, however, that the relationsnA and(nAop)�1 are di�erent in general.We will also make use of the following observation which is a consequence ofRaney's work on tight Galois connections, [Ran60].Theorem 19 (Raney) A complete lattice A is completely distributive if andonly if for every a 2 A we have a = Va0 6�aWa00 6�a0 a00.Proof. \if": It is easy to see that for every a0 6� a the element x := Wa00 6�a0 a00is completely above a. Hence Aop is super-continuous.\only if": Since a is always among the a00 of which we take the supremum inWa00 6�a0 a00, we have y := Va0 6�aWa00 6�a0 a00 � a. Assume that y is strictly above a.Then, by super-continuity, we have an element y0 completely below y but notbelow a. This y0 is one of the a0 in the formula, and it follows that y0 n y �Wa00 6�y0 a00; hence there exists a00 6� y0 which is above y0 | clearly absurd.Approximation, rather than distributivity, is used to show the following:11

Lemma 20 Let A and B be complete lattices and m:A! B be monotone.1. If A is continuous then the largest continuous function below m is givenby _m (x) = W "fm(y) j y � xg. The assignment m 7! _m is continuous asa function from the monotone function space to the continuous functionspace.2. If A is super-continuous then the largest linear function below m is given by�m (x) = W fm(y) j yn xg. The assignment m 7! �m is linear as a functionfrom the monotone function space to the linear function space.If m has �nite image within B then so do _m and �m, respectively.We need to re�ne this lemma somewhat for our purposes:Lemma 21 Let A;B be continuous lattices and letm:A! B be a _-homomorphismwhich also maps ?A to ?B. Then _m= �m.Proof. Since any supremum can be written as a combination of directed supre-mum and �nite suprema, WX = W "F��nX F , it su�ces to show that _m is still a_-homomorphism.We always have _m (a_a0) � _m (a)_ _m (a0) by monotonicity. For the converseassume b � _m (a)_ _m (a0). The set fy _ y0 j y� _m (a); y0 � _m (a0)g is directedwith supremum _m (a)_ _m (a0), so for some y � _m (a) and y0 � _m (a0) we haveb � y _ y0. The de�nition of _m gives us x � a and x0 � a0 such that y � m(x)and y0 � m(x0). Now, x _ x0 � a _ a0 and hence _m (a _ a0) � m(x _ x0) =m(x) _m(x0) � y _ y0 � b. Thus we have shown that every element way below_m (a)_ _m (a0) is also below _m (a _ a0), and so _m (a_ a0) � _m (a)_ _m (a0) followsas B is continuous.Besides approximation from below, continuous lattices also enjoy a represen-tation from above: every element x is the in�mum of ^-irreducible elements,[GHK+80, Theorem I-3.10]. If the lattice is bicontinuous then this in�mum maybe taken over the subset of ^-irreducible elements which are way-below x in Aop.In a distributive lattice there is no di�erence between ^-irreducible and ^-primeelements. Finally, an element y which is both _-prime and way-below x is actuallycompletely below x. These observations prove the following:Theorem 22 ([GHK+80]) A complete lattice is completely distributive if, andonly if, it is bicontinuous and distributive. In that case, every element is thesupremum of _-primes way-below it.Let us now put these preliminaries to work in our setting.Lemma 23 Every completely distributive lattice is an FS-lattice.12

Proof. Let A be a completely distributive lattice; it is bicontinuous by Corol-lary 18 and so every element is the supremum of _-prime elements below it. Forevery �nite subset F of _-primes de�nemF :A! A,mF (x) := W fa 2 F j a � xg.Then mF preserves �nite suprema and the conditions of Lemma 21 are satis�ed.Hence _mF is linear.Every _mF has a �nite image and so is �nitely separated from idA. The identityis equal to the directed supremum of all mF and since it itself is continuous, it isalso the directed supremum of the _mF by Lemma 20(1).Theorem 24 A complete lattice is completely distributive if and only if it is adistributive FS-lattice.Proof. This follows from Lemma 23, Corollary 5, Theorems 7 and 22.Lemma 25 The category CD of completely distributive lattices and linear mapsis closed.Proof. The lattices 2 and > are objects in CD. By the preceding theoremwe already know that the linear function space (A �� B) of two completelydistributive lattices is FS, and we only need to show distributivity. To this endobserve that the supremum of elements in (A �� B) is calculated pointwise; eventhe �nite pointwise in�mum, however, is not sup-preserving in general. Hencethe in�mum is given by Lemma 20:(f ^ g)(a) =_ ff(a0) ^ g(a0) j a0n ag :Now, given f; g; h:A �� B, we will always have (f^g)_(f^h) � f^(g_h). For theconverse �x a 2 A and assume bn (f ^ (g_h))(a). By what we just said aboutin�ma in (A �� B), there must exist a0n a such that b � f(a0)^ (g(a0)_ h(a0)).Distributivity at the element level gives us b � (f(a0) ^ g(a0)) _ (f(a0) ^ h(a0))and the latter is a term which occurs in the calculation of ((f ^ g) _ (f ^ h))(a).Theorem 26 CD is the largest closed full subcategory of SUP whose objects aredistributive and continuous.If follows that CD gives us another, smaller model of Linear Logic. Besidesits objects being even more regular than those of FS, we �nd that in CD theinterpretation of tensor and its de Morgan dual, par, coincide:Theorem 27 Let A and B be complete lattices and let one of them be completelydistributive. Then (A �� Bop) �= (Aop �� B)op, i.e. A B �= AOB.13

Proof. (Note that all operations and relation symbols in this proof refer to theoriginal lattices, not their order duals.) Given complete lattices A and B, de�ne�: (A �� Bop) ! (Aop �� B); �(r)(x) := Wx0 6�x r(x0): (Aop �� B) ! (A �� Bop); (s)(x) := Vx0 6�x s(x0) :It is clear that � and are antitone. More important is well-de�nedness:�(r)(VX) = Wx0 6�VX r(x0)= Wx2X Wx0 6�x r(x0) by the de�nition of VXand dually for . The maps � and are mutual inverses of each other. Lets:Aop �� B. Then�((s))(x) = _x0 6�x(s)(x0) = _x0 6�x ^x00 6�x0 s(x00) =: t(x) :It is clear that t(x) � s(x) because x is always one of the x00 in the formula. Forthe converse we use complete distributivity of A which entails x = Vaox a andx = Vx0 6�xWx00 6�x0 x00 (Theorem 19). Now, for ao x we get 9x0 6� x: Wx00 6�x0 x00 �a, i.e., 9x0 6� x 8x00 6� x0: x00 � a. Since s is antitone, this translates as 9x0 6�x 8x00 6� x0: s(x00) � s(a) and hence t(x) � s(a). Since s translates in�ma intosuprema, we get s(x) = s(Vaox a) = Waox s(a) � t(x).Note that we have used complete distributivity of A alone. Complete dis-tributivity of B would also su�ce since we can always switch to the other half ofa Galois-connection.In Barr's terminology, what we have shown is:Corollary 28 The category CD is compact closed.We conclude this section with an observation which is easy to justify at thispoint but will be used only in Section 7.3.Lemma 29 Let A and B be bicontinuous lattices and let F � (A �� B) be�ltered. Then the in�mum of F in (A �� B) equals the in�mum of F in [A! B].Proof. Given a �ltered family F � (A �� B) we consider the pointwise in�-mum m(x) := Vf2F f(x). It is not only monotone but also preserves the leastelement and binary suprema. This is because Bop is also continuous and on acontinuous lattice the binary in�mum is a continuous operation. Now we canapply Lemma 21 and we get that _m, which is the in�mum of m in [A! B], islinear and hence the in�mum in (A �� B).14

6 The modalitiesSo far, we have ignored the modalities of Linear Logic and it is high time to studyhow they can be added to our framework. Some general comments may be inplace here. From the viewpoint of �-autonomous categories, modalities require afurther piece of structure in the form of a comonad. First Seely, [See89], and laterBenton, Bierman, de Paiva, and Hyland, [BBHdP93, BBdPH93, Bie95], workedout the precise conditions that need to be imposed on the comonad in order to getthe desired close correspondence between proof theory and categorical semantics.More recently, Benton, [Ben94], came up with a quite di�erent notion ofcategorical model, where one has a cartesian closed category (the intuitionisticcategory) and a �-autonomous category (the linear category) linked by a monoidaladjunction. The attractions of Benton's approach are twofold: Firstly, the setof axioms is small and uses well-established concepts only. Secondly, the freeparameters in a Benton model of Linear Logic are clearly visible; neither doesthe linear category determine the intuitionistic one, nor the other way round; andonce the two categories are �xed, there may still be some variability in terms ofwhich adjunction to choose.These general bene�ts are augmented with some speci�c advantages in oursetting. Since we can choose the intuitionistic category independently from thelinear category, we have the opportunity to bring classical categories of domainsinto the picture. In other words, we are not forced to work with complete latticesalone. This ought to facilitate the application of our results to DenotationalSemantics.Although the de�nition of a Benton model is very neat, the number of dia-grams to check is still quite daunting. We are helped by the following generalresult from [Kel74] (which was also noted in [Ben94]):Theorem 30 Let (C;C ; IC) G�! (D;D; ID) F�! (C;C ; IC) be an adjunctionbetween (symmetric) monoidal categories and letn:F (A)C F (B) ��! F (AD B) p: IC ! F (ID)be a natural transformation (resp. a morphism) making the left adjoint F monoidal.Then the following are equivalent:1. The whole adjunction is monoidal.2. All arrows nA;B and p are isomorphisms.In the spirit of Denotational Semantics and Domain Theory, the natural part-ner for Barr's linear category SUP is DCPO, the category of directed-completepartial orders and Scott-continuous functions. DCPO is cartesian closed and15

is the ambient category for many of the more re�ned concepts in Domain The-ory. Our choice of adjunction is informed by our wish to decompose the mapsof DCPO. Consider the de�nitionsHD := fX � D j X Scott-closedg ;where D is a dcpo and the order on HD is subset inclusion, andiD:D! HD ; d 7! #d :(We chose the notation H because HA is almost the Hoare-powerdomain of A,except that for the latter the empty set is usually excluded.) The functions iDare Scott-continuous. Furthermore, we have the following.Lemma 31 Let D be a dcpo and B be a complete lattice. For every Scott-continuous function f :D ! B there is a unique linear function f̂ :HD �� Bsuch that f = f̂ � iD.Proof. The equality f = f̂ � iD forces the following de�nition of f̂ :f̂(X) :=_ ff(x) j #x � Xg :For linearity, let (Xi)i2I be a collection of Scott-closed subsets of D. Note thatin HD the supremum is calculated as_i2IXi = cl([i2IXi) ;where cl(�) denotes the closure of a subset in the Scott-topology. We need toshow that f̂(Wi2I Xi) � Wi2I f̂(Xi), the other inequality being satis�ed trivially.Consider the Scott-closed subset #Wi2I f̂(Xi) of B. Its pre-image under f isScott-closed by the Scott-continuity of f and contains all Xi's, hence Wi2I Xias well. So we get f(Wi2I Xi) � #Wi2I f̂(Xi) and consequently f̂(Wi2I Xi) =W ff(x) j x 2 Wi2I Xig � Wi2I f̂(Xi).From the lemma above we obtain that SUP is a re ective subcategory ofDCPO, the re ection being given byD 7! HDf :D! E 7! \iE � f :In order to show that the adjunction is monoidal we check the conditions ofTheorem 30. First of all, ISUP = 2 is clearly isomorphic to HIDCPO = H1. Weget the desired natural isomorphism between HA HB and H(A � B) from thefollowing functional description of H:HA �= [A! 2]op:16

The calculation runs as followsHA HB = (HA �� (HB)op)op�= [A! (HB)op]op�= [A! [B ! 2]]op�= [A�B ! 2]op�= H(A� B) :We also need to establish that these isomorphisms commute in a suitable waywith the transformations which correspond to the associativity, symmetry, andunit laws of the symmetric monoidal structure. For this we need a more explicitdescription of the above isomorphism.For a 2 A; b 2 B de�ne a Galois-map (a% b):A! B by(a% b) :=^ fr 2 A B j r(a) � bgor, explicitly, (a% b)(x) := 8<: >B; if x = ?A;b; if x 2 #a n f?Ag;?B; if x 62 #a:The other half of this Galois-map is just (b % a), as one can see from thecharacterization in Formula (4). Furthermore, we have r = Wa2A(a % r(a)) forall r 2 A B, because r itself is an element of the set of which the in�mum istaken in the de�nition of (a % r(a)). Also note that (?A % b) and (a % ?B)equal (?A % >B), the smallest element in A B.Using this information, we can describe the isomorphism between HA HBand H(A� B) explicitly by(#a% #b) � r () (a; b) 2 Cwhere r 2 HA HB and C 2 H(A � B). The diagrams for the monoidicity ofH:DCPO! SUP now become easy exercises. For example, commutativity ofHA HB � H(A�B)HB HAsSUP� � H(B � A)�HsDCPOis argued as follows. For r 2 HA HB we have (#a % #b) � r () (a; b) 2C () (b; a) 2 HsDCPO(C) () (#b% #a) � sSUP (r). Leaving the remainingdiagrams as exercises, we arrive at the following:17

Theorem 32 The categoriesDCPO and SUP, linked by the re ection H:DCPO!SUP, form a Benton model of Linear Logic.The theorem implies that there is a natural transformation A�B ��! AB.This, of course, is nothing other than the assignment (a; b) 7! (a% b); it is linearin both variables separately.The setup of Theorem 32 can be restricted on both sides to approximatedobjects. Since the Scott-topology of a continuous domain is a completely dis-tributive lattice, [AJ94, Theorem 7.2.28], we get a very small model by pairingScott-domains on the intuitionistic side with completely distributive lattices onthe linear side. On the other end, a maximal Benton model within approximatedordered structures is given by FS-domains paired with FS-lattices.The desired decomposition of the Scott-continuous functions space [A! B]into (HA �� B) was the motivation for our choice of the modality !A as thelattice of all Scott-closed subsets of A, ordered by set inclusion. While !A owes itsde�nition to a topological notion, the nature of ?A is then completely determinedby the structure of the ambient linear category SUP: ?A has to be naturallyisomorphic to (!Aop)op. This, in turn, is naturally isomorphic to �Aop, the Scott-topology on Aop. This works on the level of DCPO and SUP already. In theapproximated case we can give a good deal more information about ?. Recallthat a subset of a topological space is called saturated if it equals the intersectionof its neighborhoods. The set of all compact saturated subsets of a space X,ordered by revered inclusion, is denoted by �X .Proposition 33 If A is a lean complete lattice then ?A and �A are isomorphic,where the isomorphism can be viewed as the identity at the level of sets.Proof. We have remarked before that a compact upper set is necessarily closedwith respect to �Aop , that is, a member of H(Aop). The converse is exactly thede�nition of leanness.The proposition above entails that ?A �= �A holds for all FS-lattices A. Now,except for the empty set, �A is exactly the Smyth-powerdomain of A if A iscontinuous, [Smy78, AJ94]. Hence in our domain-theoretic model of Linear Logicthe two modalities are just the two fundamental powerdomains.7 AlgebraicityThe category FS has plenty of algebraic lattices as objects. Theorem 24 assures usthat FS contains at least all completely distributive algebraic lattices; moreover,every �nite lattice is certainly algebraic and FS. In this section we will explore theworld of algebraic FS-lattices in more detail. As we will see, a lot of the theoryis in close analogy to that of algebraic domains and Scott-continuous functions,but there are a few surprises. In the following, we will frequently refer to the18

classical theory of domains, so we like to alert the reader that she will �nd FS-domains next to FS-lattices and Scott-continuous functions next to linear ones inour proofs. It will be crucial that every linear function is also Scott-continuous.7.1 Algebraic FS-domainsFS-lattices are de�ned with reference to �nitely separated (linear) functions.There are two strengthenings of this concept that we will make use of here: afunction below the identity is called a de ation if it has �nite image. A de ationmay or may not be idempotent. Scott-continuous de ations are familiar from thestudy of bi�nite domains [Plo76, AJ94]; here, of course, we require them to belinear.Lemma 34 Let f be a �nitely separated function on a complete lattice A. Thensome �nite iterate of f is an idempotent de ation.Proof. The statement follows from the fact that in a sequence x > f(x) >f 2(x) > : : : a di�erent separating element is needed at least every other step.Hence such a sequence can never be longer than 2l where l is the cardinality ofthe �nite separating set. It follows that f 2l is idempotent. The iterated functionhas �nite image because it remains �nitely separated.Proposition 35 A complete lattice A is an algebraic FS-lattice if and only if theidentity idA is the directed supremum of idempotent linear de ations.Proof. \if": The image of an idempotent de ation consists wholly of compactelements. So A must be algebraic if there exists a directed family of idempotentde ations approximating idA. Since de ations are �nitely separated (by theirimage) the lattice must also be FS.\only if": Given a compact element c of A there exists a �nitely separatedfunction f which �xes c. By the previous lemma, some iterate of f is an idem-potent de ation. This iterate still �xes c. This shows that the supremum of allidempotent de ations equals idA. The supremum is directed because the point-wise supremum of idempotent de ations is another such function.This characterization of algebraic FS-lattices allows us to prove easily thatthe linear function space of two algebraic FS-lattices is again of the same kind.This observation is su�cient to conclude the following:Theorem 36 The category aFS of algebraic FS-lattices and linear maps is �-autonomous.In analogy to the Scott-continuous case, one can de�ne linear bi�nite lat-tices as the bilimits of �nite lattices with respect to linear embedding projectionpairs. The following characterization is then proved exactly as for bi�nite do-mains [Jun89, Theorem 1.26]. 19

Proposition 37 A complete lattice A is linearly bi�nite if and only if there existsa directed collection of idempotent de ations whose supremum equals idA.To summarize, what we have is:Theorem 38 For a complete lattice A the following are equivalent:1. A is an algebraic FS-lattice.2. A is linearly bi�nite.3. A has a directed collection of idempotent linear de ations whose supremumequals idA.7.2 Retracts of bi�nite latticesAs we will see in the next subsection, it is often useful to be able to pass toretracts without leaving the ambient category. We therefore collect a few basicresults about retracts of various kinds of FS-lattices.Proposition 39 The category FS is closed under retracts.Proof. For A 2 FS, B 2 SUP, let r:A �� B and e:B �� A be linear mapswith r � e = idB. If f is �nitely separated in (A �� A) by a set M , then r � f � e iseasily seen to be �nitely separated in (B �� B) by the set r(M). The rest followsfrom the local continuity of ( �� ).Corollary 40 Retracts of linear bi�nite lattices are FS-lattices.As in the Scott-continuous case, retracts of linear bi�nite lattices can be char-acterised functionally:Theorem 41 A complete lattice B is a linear retract of some linear bi�nite lat-tice if, and only if, its identity is the directed supremum of de ations in (B �� B).The question arises whether every FS-lattice is the retract of an algebraic FS-lattice (= linear bi�nite lattice). This we don't know. The situation is exactly aswith bi�nite domains and FS-domains [AJ94, Proposition 4.2.12], although wedo not see any general reason for this analogy.If we combine distributivity with algebraicity, then the problem does not arise:Theorem 42 Every distributive FS-lattice is the linear retract of a distributivelinear bi�nite lattice.Proof. A distributive FS-lattice A is automatically completely distributive byTheorem 22. Now, if A is in CD, then let B be the lattice of lower sets of_-prime elements in A ordered by inclusion. Then B is completely distributiveand algebraic. The maps r:B ! A, L 7! WL, and e:A ! B, x 7! fr j r �x; r _-primeg, are linear with r � e = idA due the Theorem 22.20

7.3 Maximality of aFSIn the case of continuous lattices, our proof techniques required lattices to belean in order to realize FS as a maximal �-autonomous subcategory of continuouslattices in SUP, Lemma 13 and Theorem 14. This topological assumption canbe eliminated in the algebraic setting [Hut95a]:Theorem 43 Let A be an algebraic lattice with continuous linear function space(A �� A). Then A is an FS-lattice.Corollary 44 aFS is the largest (full) �-autonomous subcategory of SUP suchthat every object is algebraic.The proof of the theorem above is custom-tailored for the structural propertiesof algebraic lattices; it remains unclear whether it has a suitable abstractionallowing one to prove its continuous version. We leave this as an open problem:If (A �� A) is a continuous lattice, is A necessarily lean?Since A is algebraic in the theorem above, we know that idA is the directedsupremum of idempotent, Scott-continuous de ations. Thus, it su�ces to showthat any such function d has a linear de ation p above it. We will reason theexistence of such a p in a number of steps. In the discussion below, we �x analgebraic lattice A such that (A �� A) is continuous and d is an arbitrary Scott-continuous idempotent de ation on A.Step 1: A is bicontinuous. This follows directly from Corollary 12.Step 2: Obtaining a candidate linear de ation. Any candidate linear de ationabove d has to be in the set U = ff 2 (A �� A) j d � f � idg. This set containsid and is closed under composition as composition is monotone and d and id areidempotent. The combination of these two facts establishes that U is a �lteredsubset of (A �� A) and by Lemma 29 we may conclude that its �ltered in�mum pin (A �� A) is actually the one in [A! A], using the bicontinuity of A secured inStep 1. Thus, p has to be above d. Since id is in U we get p � id. From this, theminimality of p in U , and the fact that U is closed under composition, we inferthat p is idempotent. In summary, p is the minimal idempotent linear functionabove d and below id. Since the order on such functions is given by the inclusionof their image, we conclude that there is a linear de ation above d if, and onlyif, the image of p is �nite.From now on we write B for the image of p, and i:B ! A, q:A! B for thedecomposition of p into inclusion and projection part.Step 3: (B �� B) is continuous. The pair (q; i) realizes B as a linear retract ofA. Using the internal hom ( �� ) on the pairs (q; i) and (i; q) we obtain (B �� B)as a linear retract of (A �� A). Since the Scott-continuous retract of a continuouslattice is continuous [GHK+80, AJ94], we infer that (B �� B) is continuous.21

Step 4: The identity is compact in (B �� B). The de ation d is in K[A! A] andso W = fh 2 (A �� A) j d � hg is Scott-open in (A �� A) as directed supremaare the same in [A! A] and (A �� A). Thus, p is a minimal element of theScott-open set W and the continuity of (A �� A) makes p compact in (A �� A).Using this compactness, one may now compute that q � i is compact in (B �� B),but this is just idB.Step 5: B satis�es the ascending (ACC) and descending chain condition (DCC).We already know that the identity of B is compact in (B �� B). By Lemma 11, weget that every b 2 B is compact. Since (B �� B) is isomorphic to (Bop �� Bop),we also get id 2 K(Bop �� Bop) and may use the same lemma to infer that everyb 2 B is compact in Bop. These two properties ensure that B satis�es (ACC)and (DCC).To summarize this discussion, we arrived at a bicontinuous lattice B withcontinuous linear function space (B �� B), where B satis�es (ACC) and (DCC).Let us say that any lattice C with these properties has property F. Our aim is todemonstrate that property F is nothing but that of being a �nite lattice.Step 6: Property F is inherited by principal lower and upper sets. Note thatC has property F if Cop has property F and vice versa. This is due to theisomorphism (C �� C) �= (Cop �� Cop). Thus, given C with property F, weonly have to show such a closure for a principal lower set #x. The retractionretx:C ! C which leaves #x �xed and maps all other elements to x realizes #xas a linear retract of C. As before, we obtain (#x �� #x) as a linear retract of(C �� C). In particular, (#x �� #x) is continuous. Since #x evidently inherits(ACC) and (DCC) from C, we only need to establish that #x is bicontinuous;but this follows from Corollary 12.Step 7: B is �nite. Proof by contradiction: Let us assume that B, the image ofp, is indeed in�nite. Our goal is to argue thatM1 (Example 2) is sitting inside B.Step 7.1: Finding in�nite anti-chains. Let K be a maximal chain in B. It willcontain > and ?. Since B is assumed to be in�nite, the set Y = fx 2 K j"x is in�niteg is non-empty as ? 2 Y . Since B satis�es (ACC), the set Y has amaximum m. By step 6, the lattice "m has property F. The restriction K 0 of Kto "m is a maximal chain in "m and the set Z = fx 2 K 0 j #x is in�nite in "mgis non-empty as > 2 Z. Since B satis�es (DCC), the set Z has a minimum n.By step 6, we infer that the lattice #n, the lower set of n in "m, has propertyF. Thus, we may assume, without loss of generality, that B is just this principallower set in "m. This means that1. #x is �nite for all x < > in B,2. "x is �nite for all ? < x in B.Since B satis�es (DCC), we get B n f?g = "T , where T is the set of minimalelements in B nf?g. Dually, the condition (ACC) guarantees that B nf>g = #S,22

with S being the set of maximal elements in B n f>g. Since B is in�nite, item 1implies that S is an in�nite anti-chain. Dually, item 2 implies that T is an in�niteanti-chain as well.Step 7.2: Carving out M1. We use items 1 and 2 above together with the twoin�nite anti-chains S and T to construct M1 as a linear retract of B. We de�neinductively a family of elements (xi)i�? in T and a family (Si)i�? of subsets ofS: Pick any x0 in T and de�ne S0 as "x0 \ S. By item 2 above, we see that S0 is�nite. Thus, item 1 entails that #S0 \ T is �nite as well. Since T is in�nite, wemay pick some x1 in T n #S0 and repeat this process by picking a new elementxi+1 in the complement of S1�j�i #Sj in T . Suppose that xi _xi+k < > for somei < i+ k. Then xi _ xi+k has to be below some s 2 S. Then xi � s means s 2 Siand xi+k � s renders xi+k 2 #Si contradicting the choice of the element xi+k.Thus, xi _ xj = > for all i 6= j. This ensures that fxi j i � 0g [ f?;>g is closedunder all suprema and in�ma in B and isomorphic to M1. Therefore, we havean injective map e:M1 ! B preserving all in�ma and all suprema. Becauseof the former, e has a lower adjoint l:B ! M1. The injectivity of e impliesl � e = idM1. Since lower adjoints preserve suprema, we have realized M1 as alinear retract of B. Again, this entails that (M1 �� M1) is a linear retract of(B �� B) whence (M1 �� M1) has to be continuous, contradicting Example 2.Hence the assumption that B be in�nite is false.To summarize, we have shown that there is a linear idempotent de ationabove every Scott-continuous idempotent de ation in A, and the proof that A isan FS-lattice is complete.7.4 Internal characterizationWe have seen in Sections 7.1 that algebraic FS-lattices are in fact bi�nite, andwe have characterized them in terms of idempotent de ations. So far, this isvery much in parallel to the theory of domains and Scott-continuous functions;in fact, the proofs of these facts for the linear case are virtually the same as forthe continuous case. We will now attempt to push the analogy further to theinternal characterization of bi�nite domains and lattices.Recall that bi�nite domains can be characterized by the structure of theirsubposet of compact elements [Plo81, AJ94]. Essentially, this is achieved by astudy of the �ne structure of the images of idempotent de ations. One observesthat such an image must consist of compact elements and that the image is closedunder the formation of minimal upper bounds of �nite subsets.In the present setting we will try to proceed similarly. From the continuouscase we inherit the information that the image of a linear idempotent de ationmust consist of compact elements, and consequently, the internal characterizationwill refer to compact elements only. The study of minimal upper bounds, however,is trivial for complete lattices as every subset has a supremum, and closing a �nite23

set of compact elements with all suprema will always yield a �nite set of compactelements. Hence continuous idempotent de ations abound. Our problem is toensure that there are enough linear ones.We will not study the preservation of suprema directly but instead generatea de ation together with an upper adjoint. Linearity will then be automatic. Tostart o� in this direction let us record a few observations about adjoints whichcan all be proved from the characterizing equivalence 3 in Section 3.Proposition 45 Let A be a complete lattice and f :A �� A a linear function.The following relationships hold between f and its upper adjoint f �:1. f � idA () f � � idA;2. f � f = f () f � � f � = f �;3. f has �nite image () f � has �nite image.Corollary 46 If f is a linear projection (idempotent de ation) on the completelattice A, then f � is a linear projection (idempotent de ation) on Aop.The following lemma will be the key to our characterization. It holds withoutassuming �nite image.Lemma 47 Let f be a linear projection on a complete lattice A, and let x be inim(f), the image of f . Then x creates a partition of A with the classes Ux = "xand Lx = A n "x which is respected by both f and f �, that is,f(Ux) � Ux ; f �(Ux) � Ux ;f(Lx) � Lx ; f �(Lx) � Lx :Furthermore, Lx = #f �(Lx).Proof. Assume y � x. Then f(y) � f(x) = x because f is idempotent; hencef restricts to Ux. The upper adjoint trivially restricts to Ux because we havef � � idA by Proposition 45(1) and Ux is an upper set. For the same reason, frestricts to the lower set Lx. Lastly, let y 6� x and assume f �(y) � x. Theny � f(x) by adjointness. However, f(x) = x as x belongs to the image of f andwe get a contradiction.The additional claim about Lx follows from what we just proved and the factthat f � � idA.Proposition 48 Let f be a linear projection on a complete lattice A, and letX be a subset of im(f). Then the maximal elements of LX = A n "X all belongto im(f �). 24

Proof. We have that f � restricts to LX = Tx2X Lx by the previous lemma, andthat f � is above idA by Proposition 45(1). Hence a maximal element of LX mustremain �xed under f �.This last result allows us to characterize images of linear projections.Theorem 49 The set of linear projections on a complete lattice A is in one-to-one correspondence to pairs of subsets (M;N) which have the following properties:P1. 8X �M: max(A n "X) � N ;P2. 8Y � N: min(A n #Y ) �M ;P3. 8X �M 8a 2 A n "X 9n 2 N n "X: a � n;P4. 8Y � N 8a 2 A n #Y 9m 2M n #Y: b � m.The correspondence assigns to a linear projection f the pair (im(f); im(f �)) andto a pair (M;N) the function f : a 7! W(#a \M).Proof. Given a linear projection f , then (im(f); im(f �)) has the four propertieslisted because of Lemma 47 and Proposition 48. Conversely, given a pair ofsubsets with these properties, we let f be as stated and g: a 7! V("a \N). It isclear that f is idempotent and below idA.Before we can show that f is linear, we need to establish that M is indeed allof im(f). For this, let x 2 im(f), that is x = W(#x \M). For every a 6� x theremust exist ma 2 #x \M not below a. By Property P3, there is some n 2 Nabove a and not above ma. Hence A n "x = #(N n "x). Since x is maximal inA n #(N n "x), it belongs to M by Property P2. Properties P1 and P4 are usedto show that N is all of im(g).We prove that f is linear by showing that f and g are adjoint. Assumex 6� g(y). We have just shown that g(y) 2 N and so by Property 4 thereexists m 2 M with m � x and m 6� g(y). By the de�nition of f , this entailsf(x) 6� g(y). Since y � g(y) we can't have f(x) � y. So f(x) � y impliesx � g(y). The other direction follows by duality.We had to show already that starting with a pair (M;N), constructing f fromit and taking (im(f); im(f �)) will give back (M;N). For the other identity, startwith a projection f . If follows (even in the monotone case) that f is recoveredfrom im(f) in the way stated.For projections with �nite image the characterization is even simpler:Theorem 50 Let A be a complete lattice. The set of linear idempotent de ationsis in one-to-one correspondence to pairs of �nite subsets (M;N) which have theproperties P1 and P2 from the previous theorem plusP30. M � K(A); 25

P40. N � K(Aop);The correspondence is established as before.Proof. We know from Corollary 46 that every linear idempotent de ation hasan adjoint which is a linear idempotent de ation on Aop. We also know that theimage of a linear idempotent de ation consists of compact elements only. Forthe converse we need that P30 and P40 (together with P1 and P2) imply theircounterparts in Theorem 49. This is very easy: For every X �M , the set An"Xis �A-closed by P3. Hence every element of this set is below a maximal element.The maximal elements of A n "X, however, all belong to N by P1.We need to be able to extend every �nite set M of compact elements to animage of a linear idempotent de ation, if we want that a given algebraic latticebelongs to FS. By the previous theorem, the smallest extension (if it exists) isgenerated by turning the conditions (1) and (2) into mutually dependent closureoperators: M0 := MMk+1 := SY�Nk min(A n #Y )Nk+1 := SX�Mk max(A n "X)M� := Sk2NMkN� := Sk2N NkTheorem 51 An algebraic lattice is an FS-lattice if and only if for every �nitesubset M of compact elements the sets M� and N� are �nite and consist ofcompact elements of A and Aop, respectively.It is instructive to consider in which ways the generation process can fail tolead to a linear idempotent de ation. Firstly, we observe that for a �nite set X ofcompact elements, the set "X is both open and compact. Because of the former,the complement A n "X has a maximal element above every member. The latterimplies that A n "X is open in Aop. If we assume that Aop is algebraic as well,then each maximal in A n "X is compact with respect to Aop. Hence assumingthat A is bialgebraic will guarantee that M� and N� consist of compact elementsonly.Secondly, we need that the generation process does not lead to an in�nite set.For this, we observe the following:Proposition 52 Let A be bialgebraic. Then A is lean if and only if for every Ccompact open in Aop, the set A n C is compact open in A.Proof. A set C which is compact saturated in Aop is closed in A. Hence itscomplement is open in A. As C is open in Aop, its complement is closed in Aop.The complement is then compact in A by the de�nition of leanness.For the converse, let C be closed in Aop. For every x 2 A n C there is anAop-compact element above it. Given a �nite set X of Aop-compact elements26

a�1 b�1 c�1a0 b0 c0a1 b1 c1a2 b2 c2>

?Figure 2: A bialgebraic lean lattice which is not an FS-lattice.in A n C, the set #X is compact open in Aop. By assumption, its complement(which contains C) is compact open in A. It follows that C is the �ltered in-tersection of compact open sets in A. Since algebraic lattices are sober, [AJ94,Proposition 7.2.27], C is compact as well, [AJ94, Corollary 7.2.11].As an illustration, consider the non-lean bialgebraic lattice M1 from Exam-ple 2. Here the generation process, when started on any element di�erent from> or ?, leads immediately to in�nite subsets.Unfortunately, however, leanness is not su�cient for the generation processto succeed. Figure 2 shows a bialgebraic lean lattice which is not FS. As a thirdcondition, in addition to bialgebraic and lean, we therefore need to stipulatethat the generation process terminates after �nitely many iterations. This is insurprising analogy to the classical theory of bi�nite domains. There, too, \twothirds" of being bi�nite are captured topologically (compactness of the Lawson-topology), but the remaining third is formulated with reference to a generationprocess.8 Extensions to Scott-domainsIf we drop the requirement that objects A be isomorphic to (A �� 2) �� 2, then wemay consider the category BC of bounded complete dcpos and maps f :A! Bpreserving all existing suprema: the existence of WX for X � A implies thatW f(X) exists in B and equals f(WX). Since SUP is a full subcategory of BC,we have a concrete forgetful functor with a left adjoint given by ( �� 2) �� 2[Hut95b]. This tight connection between these categories is corroborated at thelevel of objects: A embeds into (A �� 2) �� 2 such that its image is a lower27

set closed under all suprema existing in A. So while morphisms in BC do nothave an upper adjoint in general, one could de�ne the other linear types in BCusing the connections above such that the forgetful functor becomes symmetricmonoidal.Instead of providing the details, we brie y discuss the aspect of approximationin BC. If we restrict attention to continuous (Scott)-domains, then the resultingsubcategory is not closed since CL isn't. We may de�ne approximative objects Asuch that their double dual is an FS-lattice, but one may equivalently de�ne suchobjects directly as done for FS-lattices. It is not hard to see that this leads to afull symmetric monoidal closed subcategory of continuous Scott-domains in BC.One may transfer our maximality results �a la Theorems 14, 26, and 43; yet we canonly de�ne leanness indirectly by stipulating that a bounded complete continuousdomain A be \lean" if (A �� 2) �� 2 is lean in the sense we de�ned earlier. TheScott-domains obtained in this fashion were �rst introduced in [Hut94]. As fordistributivity, the domains A for which (A �� 2) �� 2 is a completely distributivealgebraic lattice are exactly Glynn Winskel's prime-algebraic domains [Win88,Hut95b].9 Related and future workIn [HM94] one �nds another, rather astonishing, external characterization of FS-lattices. Since the inclusion of (A �� B) into [A! B] is linear, it has an upperadjoint, which is just the restriction of m 7! �m to [A! B] as a domain of def-inition in Lemma 20. If A equals B and is continuous, then A is an FS-lattice(completely distributive) if, and only if, this upper adjoint is Scott-continuous(linear).In [HH98a, HH98b] one �nds a duality theory with which one can show thatthe more general continuous function space [X ! B] for a sober space X is anFS-lattice (completely distributive) if, and only if, X is a continuous space |essentially a continuous domain | and B an FS-lattice (completely distributive)[HHM99].Elements in bicontinuous lattices are in�ma of ^-irreducible elements andsuprema of _-irreducible elements. Since these elements determine the �ne-structure of such lattices, it is desirable to know whether such elements havedescriptions that re ect the type constructors, such as [!] and ( �� ), in ade-quate ways for FS-lattices. While one can use the natural isomorphism (HA ��B) �= [A! B] to arrive at such notions for the space [A! B], no identi�cationsof such elements in (A �� B) have yet been made if neither A nor B are distribu-tive. The di�culty in obtaining a characterization, say, of _-irreducible elementsin (A �� B) is linked to the open problems mentioned in this paper.28

References[AJ94] S. Abramsky and A. Jung. Domain theory. In S. Abramsky, D. M.Gabbay, and T. S. E. Maibaum, editors, Handbook of Logic in Com-puter Science, volume 3, pages 1{168. Clarendon Press, 1994.[Bar79] M. Barr. �-Autonomous Categories, volume 752 of Lecture Notes inMathematics. Springer Verlag, 1979.[Bar91] M. Barr. �-autonomous categories and linear logic. MathematicalStructures in Computer Science, 1:159{178, 1991.[BBdPH93] P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hy-land. A term calculus for intuitionistic linear logic. In M. Bezemand J. F. Groote, editors, Typed Lambda Calculi and Applications,volume 664 of Lecture Notes in Computer Science, pages 75{90.Springer Verlag, 1993.[BBHdP93] P. N. Benton, G. M. Bierman, J. M. E. Hyland, and V. C. V.de Paiva. Linear lambda calculus and categorical models revisited. InE. B�orger et al., editor, Selected Papers from Computer Science Logic'92, volume 702 of Lecture Notes in Computer Science. Springer Ver-lag, 1993.[Ben94] P. N. Benton. A mixed linear and non-linear logic: Proofs, terms andmodels. Technical Report 352, Computer Laboratory, University ofCambridge, 1994.[Bie95] G. M. Bierman. What is a model of intuitionistic linear logic?In M. Dezani-Ciancaglini and G. Plotkin, editors, Lambda Calculiand Applications, volume 902 of Lecture Notes in Computer Science,pages 78{93. Springer Verlag, 1995.[BN76] B. Banaschewski and E. Nelson. Tensor products and bimorphisms.Canad. Math. Bull., 19:385{402, 1976.[EK66] S. Eilenberg and G. M. Kelly. Closed categories. In S. Eilenberg,D. K. Harrison, S. MacLane, and H. R�ohrl, editors, Proceedings ofthe Conference on Categorical Algebra, La Jolla 1965, pages 421{562.Springer Verlag, 1966.[GHK+80] G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, andD. S. Scott. A Compendium of Continuous Lattices. Springer Verlag,1980.[Gir87] J.-Y. Girard. Linear logic. Theoretical Computer Science, 50:1{102,1987. 29

[HH98a] R. Heckmann and M. Huth. A duality theory for quantitative se-mantics. In M. Nielsen and W. Thomas, editors, Computer ScienceLogic. 11th International Workshop, volume 1414 of Lecture Notesin Computer Science, pages 255{274. Springer Verlag, 1998.[HH98b] R. Heckmann and M. Huth. Quantitative analysis, topology, andpossibility measures. Topology and its Applications, 80:1{27, 1998.[HHM99] R. Heckmann, M. Huth, and M. Mislove. Bicontinuous functionspaces. Technical Report KSU-CIS-TR-99-??, Department of Com-puting and Information Sciences. Kansas State University, January1999.[HM94] M. Huth and M. Mislove. A Characterization of linear FS-lattices.Technical Report 1679, Technische Hochschule Darmstadt, Septem-ber 1994.[Hut94] M. Huth. Linear Domains and Linear Maps. In S. Brookes, M. Main,A. Melton, M. Mislove, and D. Schmidt, editors,Mathematical Foun-dations of Programming Semantics, volume 802 of Lecture Notes inComputer Science, pages 438{453. Springer Verlag, 1994.[Hut95a] M. Huth. The greatest symmetric monoidal closed category of Scott-domains. Technical Report 1734, Technische Hochschule Darmstadt,May 1995.[Hut95b] M. Huth. A maximal monoidal closed category of distributive al-gebraic domains. Information and Computation, 116(1):pp.10{25,January 1995.[Jun89] A. Jung. Cartesian Closed Categories of Domains, volume 66 of CWITracts. Centrum voor Wiskunde en Informatica, Amsterdam, 1989.[Jun90] A. Jung. The classi�cation of continuous domains. In Logic in Com-puter Science, pages 35{40. IEEE Computer Society Press, 1990.[Kel74] G. M. Kelly. Doctrinal adjunction. In G. M. Kelly, editor, Cate-gory Seminar | Sydney 1972/73, volume 420 of Lecture Notes inMathematics, pages 257{280. Springer Verlag, 1974.[Len98] J. Leneutre. Contribution �a l'�etude de la logique lin�eairecomme formalisme de mod�elisation et de sp�eci�cation. PhD the-sis, D�epartement Informatique, �Ecole Nationale Superieure desT�el�ecommunications, Paris, 1998.[Mac71] S. Mac Lane. Categories for the Working Mathematician. SpringerVerlag, 1971. 30

[Plo76] G. D. Plotkin. A powerdomain construction. SIAM Journal onComputing, 5:452{487, 1976.[Plo81] G. D. Plotkin. Post-graduate lecture notes in advanced domain the-ory (incorporating the \Pisa Notes"). Dept. of Computer Science,Univ. of Edinburgh, 1981.[Ran53] G. N. Raney. A subdirect-union representation for completely dis-tributive complete lattices. Proc. AMS, 4:518{522, 1953.[Ran60] G. N. Raney. Tight Galois connections and complete distributivity.Trans. AMS, 97:418{426, 1960.[Sco72] D. S. Scott. Continuous lattices. In E. Lawvere, editor, Toposes,Algebraic Geometry and Logic, volume 274 of Lecture Notes in Math-ematics, pages 97{136. Springer Verlag, 1972.[See89] R. Seely. �-autonomous categories, cofree coalgebras and linear logic.In J. W. Grey and A. Scedrov, editors, Categories in Computer Sci-ence and Logic, volume 92 of Contemporary Mathematics, pages 371{382. American Mathematical Society, 1989.[Smy78] M. B. Smyth. Powerdomains. Journal of Computer and SystemsSciences, 16:23{36, 1978.[Smy83] M. B. Smyth. The largest cartesian closed category of domains.Theoretical Computer Science, 27:109{119, 1983.[Win88] G. Winskel. An introduction to event structures. In J. W. de Bakker,editor, Linear Time, Branching Time, and Partial Order in Logicsand Models for Concurrency, volume 354 of Lecture Notes in Com-puter Science, pages 364{399. Springer Verlag, 1988.Michael HuthDepartment of Computing and In-formation SciencesKansas State University234 Nichols HallManhattan, KS 66506USAhuth@cis.ksu.eduAchim JungSchool of Computer ScienceThe University of BirminghamEdgbastonBirmingham, B15 2TTEnglandA.Jung@cs.bham.ac.uk

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Klaus KeimelFachbereich MathematikTechnische Hochschule DarmstadtSchlo�gartenstra�e 764289 DarmstadtGermanykeimel@mathematik.tu-darmstadt.de

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