Natural dualities through product representations: bilattices and beyond

This paper focuses on natural dualities for varieties of bilattice-based algebras.Such varieties have been widely studied as semantic models in situations where information is incomplete or inconsistent. The most popular tool for studying bilattices-based algebras is product representation. The authors recently set up a widely applicable algebraic framework which enabled product representations over a base variety to be derived in a uniform and categorical manner. By combining this methodology with that of natural duality theory, we demonstrate how to build a natural duality for any bilattice-based variety which has a suitable product representation over a dualisable base variety. This procedure allows us systematically to present economical natural dualities for many bilattice-based varieties, for most of which no dual representation has previously been given. Among our results we highlight that for bilattices with a generalised conflation operation (not assumed to be an involution or commute with negation). Here both the associated product representation and the duality are new. Finally we outline analogous procedures for pre-bilattice-based algebras (so negation is absent).


Introduction
Bilattices, with and without additional operations, have been identified by researchers in artificial intelligence and in philosophical logic as of value for analysing scenarios in which information may be incomplete or inconsistent. Over twenty years, a bewildering array of different mathematical models has been developed which employ bilattice-based algebras in such situations; [19,23,15,26] give just a sample of the literature. Within a logical context, bilattices have been used to interpret truth values of formal systems. The range of possibilities is illustrated by [2,1,17,18,16,5,27,25].
To date, the structure theory of bilattices has had two main strands: product representations (see in particular [4,11,9] and references therein) and topological duality theory [24,22,8]. In this paper we entwine these two strands, demonstrating how a dual representation and a product representation can be expected to fit together and to operate in a symbiotic way. Our work on distributive bilattices in [8] provides a prototype. Crucially, as in [8], we exploit the theory of natural dualities; see Section 3.
In [9] we set up a uniform framework for product representation. We introduced a formal definition of duplication of a base variety of algebras which gives rise to a new variety with additional operations built by combining suitable algebraic terms in the base language and coordinate manipulation (details are recalled in Section 2). This construction led to a very general categorical theorem on product representation [9,Theorem 3.2] which makes overt the intrinsic structure of such representations. The examples we present below all involve bilattice-based varieties, but we stress that 2010 Mathematics Subject Classification. primary: 08C20, secondary 03G10, O3G25, 06B10, 06D50.
the scope of the theorem is not confined to such varieties. Our Duality Transfer Theorem (Theorem 3.1) demonstrates how a natural duality for a given base class immediately yields a natural duality for any duplicate of that class. Moreover, the dualities for duplicated varieties mirror those for the base varieties, as regards both advantageous properties and complexity (note the concluding remarks in Section 4). By combining the Duality Transfer Theorem with product representation we can set up dualities for assorted bilattice-based varieties (see Section 4, Table 1). In almost all cases the dualities are new. The varieties in question arise as duplicates of B (Boolean algebras), D (bounded distributive lattices) K (Kleene algebras), DM (De Morgan algebras), and DB (bounded distributive bilattices), all of which have amenable natural dualities (see [10] and also [8]). Variants are available when lattice bounds are omitted.
We contrast key features of our natural duality approach with earlier work on dualities for bilattice-based algebras. We stress that our methods lead directly to dual representations which are categorical: morphisms do not have to be treated case-by-case as an overlay to an object representation (as is done in [24,22]). Others' work on dualities in the context of distributive bilattices has sought instead, for a chosen class of algebras, a dual category which is an enrichment of a subcategory of Priestley spaces, that is, they start from Priestley duality, applied to the distributive lattice reducts of their algebras, and then superimpose extra structure to capture the suppressed operations. This strategy has been successfully applied to very many classes of distributive-lattice-based algebras, but it has drawbacks. Although the underlying Priestley duality is natural, the enriched Priestley space representation rarely is. Accordingly one cannot expect the rewards a natural duality offers, such as instant access to free algebras.
Section 5 focuses on the variety DB´of (bounded) distributive bilattices with a conflation operation´which is not assumed to be an involution or to commute with the negation. This variety has not been investigated before and would not have susceptible to earlier methods. We realise DB´as a duplicate of the variety DO of double Ockham algebras and set up a natural duality for DO, whence we obtain a duality for DB´. Both results are new. This example is also a novelty within bilattice theory since it takes us outside the realm of finitely generated varieties without losing the benefits of having a natural duality.
In Section 6 we consider the negation-free setting of pre-bilattice-based algebras, and link the ideas of [9, Section 9] with dual representations. Again, a very general theorem enables us to transfer a known duality from a base variety to a suitably constructed duplicate. Here multisorted duality theory is needed. Nonetheless the ideas and the categorical arguments are simple, and the proof of Theorem 3.1 is easily adapted.

The general product representation theorem recalled
We shall assume that readers are familiar with the basic notions concerning bilattices. A summary can be found, for example, in [4] and a bare minimum in [9, Section 2]. Here we simply draw attention to some salient points concerning notation and terminology since usage in the literature varies. Except in Section 6 we assume that a negation operator is present.
A (unbounded) bilattice is an algebra A " pA; _ t ,^t, _ k ,^k, q, where the reducts A t :" pA; _ t ,^tq and A k :" pA; _ k ,^kq are lattices (respectively the truth lattice and knowledge lattice). The operation , capturing negation, is an endomorphism of A k and a dual endomorphism of A t .
Bilattice models come in two flavours: with and without bounds. Which flavour is preferred (or appropriate) may depend on an intended application, or on mathematical considerations. We refer to [8,Section 1] for the formal definition of the terms bounded and unbounded. Here we merely issue a reminder that when universal bounds for the lattice order are not included in the algebraic language for a class of lattice-based algebras then the algebras involved may, but need not, have bounds; when bounds do exist these do not have to be preserved by homomorphisms. A subscript u on the symbol denoting a category will indicate that we are working in the unbounded setting. So, for example, D denotes the category of bounded distributive lattices and D u the category of all distributive lattices.
All the bilattices considered in this paper are distributive, meaning that each of the four lattice operations distributes over each of the other three. The weaker condition of interlacing is necessary and sufficient for a bilattice to have a product representation. However varieties of interlaced bilattice-based algebras seldom come within the scope of natural duality theory.
Our investigations involve classes of algebras, viewed both algebraically and categorically. We draw, lightly, on some of the basic formalism and theory of universal algebra, specifically regarding varieties (alias equational classes) and prevarieties; a standard reference for this material is [6]. A class of algebras over a common language will be regarded as a category in the usual way: the morphisms are all the homomorphisms. The variety generated by a family M of algebras of common type is denoted VpMq. Equivalently VpMq is the class HSPpMq of homomorphic images of subalgebras of products of algebras in M. The prevariety generated by M is the class ISPpMq whose members are isomorphic images of subalgebras of products of members of M. Usually the algebras in M will be finite.
We now recall our general product representation framework [9, Section 3]. We fix an arbitrary algebraic language Σ and let N be a family of Σ-algebras. Let Γ be a set of pairs of Σ-terms such that, for pt 1 , t 2 q P Γ, the terms t 1 and t 2 have common even arity, denoted 2n pt1,t2q . We view Γ as an algebraic language for a family of algebras P Γ pNq (N P N ), where the arity of pt 1 , t 2 q P Γ is n pt1,t2q . We write rt 1 , t 2 s when the pair pt 1 , t 2 q is regarded as belonging to Γ, qua language. For A P VpN q we define a Γ-algebra P Γ pAq " pAˆA; trt 1 , t 2 s PΓpAq | pt 1 , t 2 q P Γuq, in which the operation rt 1 , t 2 s PΓpAq : pAˆAq n Ñ AˆA is given by rt 1 , t 2 s PΓpAq ppa 1 , b 1 q, . . . , pa n , b n qq " pt A 1 pa 1 , b 1 , . . . , a n , b n q, t A 2 pa 1 , b 1 , . . . , a n , b n qq, where n " n pt1,t2q and pa 1 , b 1 q, . . . , pa n , b n q P AˆA. It is easy to check that the assignment A Þ Ñ P Γ pAq (on objects) and h Þ Ñ hˆh (on morphisms) defines a functor P Γ : VpN q Ñ VpP Γ pN qq. We shall also need the following notation. Given a set X the map δ X : X Ñ XˆX is given by δ X pxq " px, xq and π X 1 , π X 2 : XˆX Ñ X denote the projection maps.
We are ready to recall a key definition from [9, Section 3], where further details can be found. We say that Γ duplicates N and that A " VpP Γ pN qq is a duplicate of B if the following conditions on N and Γ are satisfied: (L) for each n-ary operation symbol f P Σ and each i P t1, 2u there exists an n-ary Γ-term t (depending on f and i) such that π N i˝t PΓpNq˝p δ N q n " f N for each N P N ; (M) there exists a binary Γ-term v such that v PΓpNq ppa, bq, pc, dqq " pa, dq for N P N and a, b P N ; (P) there exists a unary Γ-term s such that s PΓpNq pa, bq " pb, aq for N P N and a, b P N .
We now present the Product Representation Theorem [9, Theorem 3.2].
Theorem 2.1. Assume that Γ duplicates a class of algebras N and let B " VpN q.
Then the functor P Γ : B Ñ A sets up a categorical equivalence between B and its duplicate A " VpP Γ pN qq.
The classes of algebras arising in this section have prinicipally been varieties. In the next section we concentrate on singly-generated prevarieties. The following corollary tells us how the class operators HSP and ISP behave with respect to duplication. It is an almost immediate consequence of the fact that P Γ is a categorical equivalence; assertion (c) follows directly from (a) and (b).
Corollary 2.2. Assume that Γ duplicates a class of algebras M. The following statements hold for each A P VpMq: (a) HSPpP Γ pAqq is categorically equivalent to HSPpAq.

Natural duality and product representation
It is appropriate to recall only in brief the theory of natural dualities as we shall employ it. A textbook treatment is given in [10] and a summary geared to applications to distributive bilattices in [8,Sections 3 and 5].
Our object of study in this section will be a prevariety A generated by an algebra M, so that A " ISPpMq. (Only in Section 6 will we replace the single algebra M by a family of algebras M. We shall then need to bring multisorted duality theory into play.) Traditionally (and in [10] in particular) M is assumed to be finite. This suffices for our applications in Section 4. However our application to bilattices with generalised conflation will depend on the more general theory presented in [12]. Therefore we shall assume that M can be equipped with a compact Hausdorff topology T with respect to which it becomes a topological algebra. When M is finite T is necessarily discrete.
Our aim is to find a second category X whose objects are topological structures of common type and which is dually equivalent to A via functors D : A Ñ X and E : X Ñ A. Moreover-and this is a key feature of a natural duality-we want each algebra A in A to be concretely representable as an algebra of continuous structure-preserving maps from DpAq (the dual space of A) into M " , where M " P X has the same underlying set M as does M. For this to succeed, some compatibility between the structures M and M " will be necessary. We consider a topological structure M " " pM ; G, R, Tq where ‚ T is a topology on M (as demanded above); ‚ G is a set of operations on M , meaning that, for g P G of arity n ě 1, the map g : M n Ñ M is a continuous homomorphism (any nullary operation in G will be identified with a constant in the type of M); ‚ R is a set of relations on M such that if r P R is n-ary (n ě 1) then r is the universe of a topologically closed subalgebra r of M n . We refer to such a topological structure M " as an alter ego for M and say that M " and M are compatible. Of course. the topological conditions imposed on G and R are trivially satisfied if M is finite. (The general theory in [10] allows an alter ego also to include partial operations, but they do not arise in our intended applications.) We use M " to build a new category X. We first consider structures of the same type as M " . These have the form X " pX; G X , R X , T X q where T X is a compact Hausdorff topology and G X and R X are sets of operations and relations on X in bijective correspondence with those in G and R, with matching arities. Isomorphisms between such structures are defined in the obvious way. For any non-empty set S we give M S the product topology and lift the elements of G and R pointwise to M S . The topological prevariety generated by M " is X :" IS c P`pM " q, the class of isomorphic copies of closed substructures of non-empty powers of M " , with`indicating that the empty structure is included. We make X into a category by taking all continuous structure-preserving maps as the morphisms.
As a consequence of the compatibility of M " and M, and the topological conditions imposed, the following assertions are true. Let A P A and X P X. Then ApA, Mq may be seen as a closed substrucructure of M " A and XpX, M " q as a subalgebra of M X . We can set up well-defined contravariant hom-functors D : A Ñ X and E : X T Ñ A; on objects: D : A Þ Ñ ApA, Mq, on morphisms: D : x Þ Ñ´˝x, and on objects: The following assertions are part of the standard framework of natural duality theory. Details can be found in [10, Chapter 2]; see also [12,Section 2]. Given A P A and X P X, we have natural evaluation maps e A : a Þ Ñ´˝a and ε X : x Þ Ñ´˝x, with e A : A Ñ EDpAq and ε X : X Ñ DEpXq. Moreover pD, E, e, εq is a dual adjunction. Each of the maps e A and ε X is an embedding. We say that M " yields a duality on A, or simply that M " dualises M, if each e A is surjective, so that it is an isomorphism e A : A -EDpAq. A dualising alter ego M " plays a special role in the duality it sets up: it is the dual space of the free algebra on one generator in A. This fact is a consequence of compatibility. More generally, the free algebra generated by a non-empty set S has dual space M " S . Assume that M " yields a duality on A and in addition that each ε X is surjective and so an isomorphism. Then we say M " fully dualises M or that the duality yielded by M " is full. In this case A and X are dually equivalent. Full dualities are particularly amenable if they are strong; this is the requirement that the alter ego be injective in the topological prevariety it generates. We do not need here to go deeply into the topic of strong dualities (see [10, Chapter 3] for a full discussion) but we do note in passing that each of the functors D and E in a strong duality interchanges embeddings and surjections-a major virtue if a duality is to be used to transfer algebraic problems into a dual setting.
We are ready to present our duality theorem for duplicated (pre)varieties. Our notation is chosen to match that in Theorem 2.1.
Theorem 3.1 (Duality Transfer Theorem). Let N be an algebra and assume that Γ duplicates N. If the topological structure N " " pN ; G, R, Tq yields a duality on B " ISPpNq with dual category Y " IS c P`pN " q, then N " 2 yields a duality on A " ISPpP Γ pNqq, again with Y as the dual category. If the former duality is full, respectively strong, then the same is true of the latter.
Proof. For the purposes of the proof we shall assume that N , and hence also M , is finite. It is routine to check that the topological conditions which come into play when N is infinite lift to the duplicated set-up.
We claim that N " 2 acts as a legitimate alter ego for M :" P Γ pNq. Certainly these structures have the same universe, namely NˆN . It follows from the definition of the operations of P Γ pNq that P Γ prq, whose universe is rˆr, is a subalgebra of pP Γ pNqq n whenever r P R is the universe of a subalgebra r of N n . But R N " 2 consists of the relations rˆr, for r P R. Likewise, an n-ary operation g in G gives rise to the same operation, viz. gˆg, of P Γ pNq and in the structure N " 2 . Hence gˆg is compatible with P Γ pNq.
We now set up the functors for the existing duality for ISPpNq and for the duality sought for ISPpMq.
Since Y " X, the functors D B and D A have a common codomain.
Let A P A. By Corollary 2.2, we may assume that A " P Γ pBq, for some B P B. By Theorem 2.1 and the definition of P Γ on morphisms, This proves that e A : A Ñ E A D A pAq is surjective for each A P A, so that we do indeed have a duality for A based on the alter ego M " " N " 2 . We now claim that if N " fully dualises N then M " fully dualises M. To do this we shall show that the bijection η : D B pBq Ñ D A pAq, defined by ηpyq " yˆy for each y P D B pBq, is an isomorphism (of topological structures) from D B pBq onto D A pAq, where, as before, A " P Γ pBq, see [10, Lemma 3.1.1]. Let r be an n-ary relation in N " . For y 1 , . . . , y n P D B pBq, py 1 , . . . ,y n q P r D B pBq ðñ @a P N ppy 1 paq, . . . , y n paqq P rq ðñ @pa 1 , a 2 q P M pppy 1 pa 1 q, y 1 pa 2 qq, . . . , py n pa 1 q, y n pa 2 qqq P rˆrq ðñ py 1ˆy1 , . . . , y nˆyn q P prˆrq D A pAq .
A similar argument applies to operations.
The map η has compact codomain and Hausdorff domain and hence is a homeomorphism provided η´1 is continuous. To prove this it will suffice to show that each map π b˝η´1 is continuous, where π b denotes the projection from D B pBq, regarded as a subspace of N " This proves the continuity assertion.
Finally, since N " is injective in Y if and only if N " 2 is, N " yields a strong duality on B if and only if N " 2 yields a strong duality on A, by [10, Theorem 3.2.4].
The proof of Theorem 3.1 is essentially routine, given the Product Representation Theorem. The theorem should not be disparaged because it is easy to derive. Rather the reverse: almost all the dualities given in Section 4 are new, and obtained at a stroke.
Of course, though, Theorem 3.1 is only useful when we have a (strong) duality to hand for the base class ISPpNq we wish to employ. Nothing we have said about natural dualities so far tells us how to find an alter ego N " for N, or even whether a duality exists. Fortunately, simple and well-understood strong dualities exist for the base varieties ISPpNq which support the miscellany of logic-oriented examples presented in Section 4. In all cases considered there, N is a small finite algebra with a lattice reduct. Existence of such a reduct guarantees dualisability [10, Section 3.4]: a brute-force alter ego N " " pN ; SpN 2 q, Tq is available. However this default choice is likely to yield a tractable duality only when N is very small. Otherwise the subalgebra lattice SpN 2 q is generally unwieldy. Methodology exists for slimming down a given dualising alter ego to yield a potentially more workable duality (see [10,Chapter 8]), but it is preferable to obtain an economical duality from the outset. This is often possible when N is a distributive lattice, not necessarily finite: in many such cases one can apply the piggyback method which originated with Davey and Werner (see [10,Chapter 7] and [12]). We shall demonstrate its use in Section 5, where we develop a duality for double Ockham algebras, our base variety for studying generalised conflation.
Against this background we can appreciate the merits of Theorem 3.1. Suppose we have a class ISPpMq (with M finite) which is expressible as a duplicate of a dualisable base variety ISPpNq. Then |M | " |N | 2 and, on cardinality grounds alone, finding an amenable duality directly for ISPpMq could be challenging, whereas the chances are much higher that we have available, or are able to set up, a simple dualising alter ego N " for N. And then, given N " we can immediately obtain an alter ego M " for M, with the same number of relations and operations in M " as in N " .

Examples of natural dualities via duplication
We now present a miscellany of examples. All involve bilattices but, as noted earlier, the scope of our methods is potentially wider. We derive (strong) dualities for certain (finitely generated) duplicated varieties given in [9] by calling on wellknown (strong) dualities for their base varieties. A catalogue of base varieties and duplicates is assembled in [9, Appendix, Table 1], with references to where in the paper these examples are presented. Table 1 lists alter egos for dualities for base varieties. These dualities are discussed in [10], with their sources attributed. Natural dualities for the indicated duplicated varieties, also strong, can be read off from the table, using the Duality Transfer Theorem. When specifying a generator for each base variety, we adopt abbreviations for standard sets of operations: we have elected to denote negation in Boolean algebras, De Morgan algebras and Kleene algebras by ", to distinguish it from bilattice negation, . The top row of Table 1 should be treated as a prototype, both algebraically and dually. There the base variety is D, the variety of bounded distributive lattices. The duplicated variety in this case is the variety DB of distributive bilattices. It is generated (as a prevariety) by the four-element algebra in DB. Full details of the natural duality for DB and its relationship to Priestley duality for the base variety D appear in [8]. All  This is the situation with negation-by-failure. For the natural dualities recorded in Table 1, we note that, apart from D, the base variety in each case is De Morgan algebras or a subvariety thereof. The alter ego includes a partial order ď known as the alternating order in [10, Theorem 4.3.16]; in the case of DM, the relation ď on universe t0, 1u 2 of the four-element generator 4 DM is the knowledge order. The map g is the involution swapping the coordinates.
Only simple modifications are needed to handle the case when the language of a lattice-based variety does not include lattice bounds as nullary operations. It is an old result that Priestley duality for the variety D u can be set up in much the same way as that for D, with the dual category being pointed Priestley spaces, as described in [10, Section 1.2 and Subsection 4.3.1]. Natural dualities for duplicates of D u are derived from those for corresponding duplicates of D simply by adding to the alter ego nullary operations p0, 0q and p1, 1q. Compare with [8,Section 4], which provides a direct treatment of duality for DB u ; here, even more than in the bounded case, we see the merit of the automatic process that Theorem 3.1 supplies. A duality for DM u (De Morgan lattices) is obtained by adding the top and bottom elements for the partial order ď to the alter ego for DM. Our transfer theorem then applies to unbounded distributive bilattices with conflation.

Bilattices with generalised conflation
In this section we break new ground, both in relation to product representation and in relation to natural duality.
The bilattice-based variety DB´that we study-(bounded) distributive bilattices with generalised conflation-has not been considered before. Previous authors who have studied product representation when conflation is present have assumed that this operation is an involution that commutes with negation (see [14,Theorem 8.3], [4] and our treatment in [9, Section 5]). We shall demonstrate that neither assumption is necessary for the existence of a product representation.
Our focus in this paper is on developing theoretical tools. Nevertheless we should supply application-oriented reasons to justify investigating generalised conflation. We first note that it is often, but not always, natural to assume that conflation be an involution. On the other hand, the justification for the commutation condition is less clear cut. Indeed, both the original definition in [14] and that in [25] exclude commutation, and this is brought in only later. In [25, Section 3] the emphasis is on truth values. The authors' desired interpretation then leads them to consider a special algebra SIXTEEN 3 , in which the conflation operation does commute with negation. In [18,Section 2] conflation is used to study (knowledge) consistent and exact elements of a lattice. The investigations in both [25] and [18] are intrinsically connected to the product representation for bilattices with conflation. Our product representation would permit similar interpretations when commutation fails and/or conflation is not an involution. In a different setting, conflation has been used in [15] to present an algebraic model of the logic system of revisions in databases, knowledge bases, and belief sets introduced in [23]. In this model the coordinates of a pair in a product representation of a bilattice are interpreted as the degrees of confidence for including in a database an item of information and for excluding it. Conflation then models the transformation of information that reinterprets as evidence for inclusion whatever did not previously count as evidence against, and vice versa. That is, conflation comprises two processes: given the information against (for) a certain argument, these capture information for (against) the same argument. In [15] these two transformations coincide, and are mutually inverse. Our work on generalised conflation would allow these assumptions to be weakened so facilitating a wider range of models.
The class DB´consists of algebras of the form A " pA; _ t ,^t, _ k ,^k, ,´, 0, 1q, where the reduct of A obtained by suppressing´belongs to DB and´is an endomorphism of A t and a dual endomorphism of A k . Here we elect to include bounds. The variety DBC of (bounded) distributive bilattices with conflation (where by convention conflation and negation do commute) is a subvariety of DB´. However DB´and DBC behave quite differently: even though is an involution, is not. As a consequence the monoid these operations generate is not finite, as is the case in DBC. (We note that the unbounded case of generalised conflation could also be treated by making appropriate modifications to the above definition and throughout what follows.) Our product representation for DB´uses as its base variety the class DO of double Ockham algebras. This is a new departure as regards representations of bilattice expansions. A double Ockham algebra is a D-based algebras equipped with two dual endomorphisms of the D-reducts. An Ockham algebra carries just one such operation. The variety O of Ockham algebras, which includes Boolean algebras, De Morgan algebras and Kleene algebras among its subvarieties, has been exhaustively studied, both algebraically and via duality methods, as indicated by the texts [3,10] and many articles. The variety DO is much less well explored. The remainder of the section is accordingly organised as follows. Proposition 5.1 presents the product representation for DB´over the base variety DO. We then set DB´aside while we develop the theory of DO which we need if we are to apply our Duality Transfer Theorem to DB´. This requires us first to identify an algebra M such that DO " ISPpMq (Proposition 5.2). We then set up an alter ego M " for M and call on [12,Theorem 4.4] to obtain a natural duality for DO (Theorem 5.6). This is then combined with Theorem 3.1 to arrive at a natural duality for DB´(Theorem 5.7).
To motivate how we can realise DB´as a duplicate of DO we briefly recall from [9, Section 5] how DBC arises as a duplicate of DM. We adopt the notation introduced in [9, Section 4]. Let Σ be a language and f be an n-ary function symbol in Σ. For m ě n and i 1 , . . . , i n P t1, . . . , mu we denote by f m i1¨¨¨in the mary term f m i1...in px 1 , . . . , x m q " f px i1 , . . . , x in q. We can capture the extra operatioń on the generator 16 DBC of DBC using the De Morgan negation ", combined with coordinate-flipping: the family of terms Γ DBC " Γ DB Y tp" 2 2 , " 2 1 qu acts as a duplicator for DM with DBC as the duplicated variety; here Γ DB duplicates bounded lattices. (See [9, Section 5] for an explanation as to why the form of the operations in DBC dictates that DM should be used as the base variety.) We now present our duplication result linking DO and DB´.
Proposition 5.1. The set Γ DB´" Γ DB Y tpf 2 2 , g 2 1 qu duplicates DO. Moreover, DB´" V`P Γ DB´p DOq˘, where Σ Γ DB´i s identified with the language of DB´.
Proof. Certainly Γ DB´d uplicates DO because pf 2 2 , g 2 1 q P Γ DB´a nd Γ DB is a duplicate for Σ D on D.
This theorem gives insight into the effect of reinstating the assumptions customarily imposed on conflation and which we removed in passing from DBC to DB´. From the product representation for DB´, it follows that´is involutive if and only if f and g are. The resulting subvariety of DB´is a duplicate of double De Morgan algebras (that is, algebras in DO such that both unary operations are involutions). Similarly,´commutes with if and only if f " g. This time we obtain a subvariety of DB´which duplicates O.
We now want to identify an (infinite) algebra which generates our base variety DO as a prevariety. We take our cue from the variety O of Ockham algebras: O is generated as a prevariety by an algebra M whose universe is t0, 1u N0 , where N 0 " t0, 1, 2. . . .u; lattice operations and constants are obtained pointwise from the two-element bounded lattice and, identifying the elements as infinite binary strings, negation is given by a left shift followed by pointwise Boolean complementation on t0, 1u. See for example [12,Section 4] for details. We may view the exponent N 0 as the free monoid on one generator e, with 0 as identity and n acting as the n-fold composite of e.
For DO, analogously, we first consider the free monoid E " te 1 , e 2 u˚on two generators e 1 and e 2 and identify it with the set of all finite words in the language with e 1 and e 2 as function symbols, with the empty word corresponding to the identity element 1; the monoid operation¨is given by concatenation. For s P E, we denote the length of s by |s|.
For us, DO will serve as a base variety. Accordingly we align our notation with that in Theorem 3.1. We now consider the algebra N with universe t0, 1u E with lattice operations and constants given pointwise. The lattice t0, 1u E is in fact a Boolean lattice, whose complementation operation we denote by c. The dual endomorphisms f and g are given as follows. For a P t0, 1u E we have f paqpsq " cpaps¨e 1 qq and gpaq " cpaps¨e 2 qq for every s P E. This gives us an algebra N :" pt0, 1u E ; _,^, f, g, 0, 1q P DO.
For future use we show how to assign to each word s P E a unary term t s in the language of DO, as follows. If s " 1 (the empty word) then t s is the identity map; if s " e 1¨s 1 then t s " f˝t s 1 ; and if s " e 2¨s 1 then t s " g˝t s 1 . Structural induction shows that the term function t N s is given by for every a P N and s P E. # xpt s pcqq if |s| is even, 1´xpt s pcqq if |s| is odd, for c P A and s P E. It is routine to check that ϕ is a D-morphism which preserves f and g. Finally, ϕpcqp1q " xpcq, whence ϕpaq ‰ ϕpbq.
We now seek a natural duality for DO which parallels that which is already known for the category O of Ockham algebras. Our treatment follows the same lines as that given for O in [12,Section 4], whereby a powerful version of the piggyback method is deployed. (The duality for O was originally developed by Goldberg [21] and re-derived as an early example of a piggyback duality by Davey and Werner [13].) A general description of the piggybacking method and the ideas underlying it can be found in [12,Section 3]. We wish to apply to DO a special case of [12,Theorem 4.4]. We first make some comments and establish notation. We piggyback over Priestley duality between D " ISPp2q and P " IS c P`p2 " q (where 2 and 2 " are the two-element objects in D and P with universe t0, 1u, defined in the usual way). We denote the hom-functors setting up the dual equivalence between D and P by H and K. The aim is to find an element ω P DpN 5 , 2q which, together with endomorphisms of N, captures enough information to build an alter ego N " of N which yields a full duality, in fact, a strong duality.
We now work towards showing that we can apply [12,Theorem 4.4] to DO " ISPpNq, where N is as defined above. We shall take ω : N Ñ 2 to be the projection map given by ωpaq " ap1q. We want to set up an alter ego N " " pt0, 1u E ; G, R, Tq so that in particular N " has a Priestley space reduct N " 5 such that ω P PpN " 5 , 2 " q. Moreover we need the structure N " to be chosen in such a way that the conditions (1)-(3) in [12,Theorem 4.4] are satisfied. We define T to be the product topology on N " t0, 1u E derived from the discrete topology on t0, 1u; this is compact and Hausdorff and makes N into a topological algebra. We now need to specify G and R. We would expect R to contain an order relation ď such that pt0, 1u E ; ď, Tq P P. For Ockham algebras-where one uses the free monoid on one generator as the exponent rather than E-the corresponding order relation is the alternating order in which alternate coordinates are order-flipped; see [10, Section 7.5] (and recall the comment about De Morgan algebras, a subvariety of O, in Section 4). The key point is that a composition of an even (respectively odd) number of order-preserving selfmaps on an ordered set is order-preserving (respectively order-reversing). Hence the definition of ď in Lemma 5.3 is entirely natural. Lemma 5.3. Let N be as above. Then ď, given by is an order relation making pt0, 1u E ; ď, Tq a Priestley space. Moreover ď is the universe of a subalgebra of N 2 and this subalgebra is the unique maximal subalgebra of pω, ωq´1pďq " t pa, bq P N 2 | ωpaq ď ωpbq u.
Proof. Each of 2 " and the structure 2 " B (that is, 2 " with the order reversed) is a Priestley space. It follows that the topological structure pt0, 1u E ; ď, Tq is a product of Priestley spaces and so itself a Priestley space.
Take a, b, c, d in N such that a ď b and c ď d and let s P E. Then pa^cqpsq " apsq^cpsq ď bpsq^dpsq " pb^dqpsq if |s| is even, pa^cqpsq " apsq^cpsq ě bpsq^dpsq " pb^dqpsq if |s| is odd.
Likewise gpaq ď gpbq. Thus ď is indeed the universe of a subalgebra of N 2 . Now let r be the universe of a subalgebra of N 2 maximal with respect to inclusion in pω, ωq´1pďq. Then, with t s as defined earlier for s P E, we have pa, bq P r ùñ p@s P Eq`pt s paq, t s pbqq P r˘ùñ p@s P Eq`t s paq ď t s pbqù ñ p@s P Eqp@e P Eq`t s paqpeq " 1 ùñ t s pbqpeq " 1˘. But We deduce that r is a subset of ď. In addition a ď b implies ωpaq ď ωpbq: consider s " 1. Maximality of r implies that r equals ď. Consequently ď is the unique maximal subalgebra contained in pω, ωq´1pďq.
We now introduce the operations we shall include in our alter ego N " . Let the map γ i : E Ñ E be given by γ i psq " s¨e i . Then we can define an endomorphism u i of N by u i paq " a˝γ i , for i " 1, 2. These maps are continuous with respect to the topology T we have put on N . We define N " :" pt0, 1u E ; u 1 , u 2 , ď, Tq.
Then N " is compatible with N. We let Y :" IS c P`pN " q be the topological prevariety generated by N " and by 5 the forgetful functor from Y into P which suppresses the operations u 1 and u 2 . We note that now ω, as defined earlier, may be seen to belong to DpN 5 , 2q X PpN " 5 , 2 " q. The following two lemmas concern the interaction of N, N " and ω as regards separation properties.
Lemma 5.4. Assume that N, N " and ω are defined as above. Then, given a ‰ b in N , there exists a unary term u in the language of pN ; u 1 , u 2 q such that ωpupaqq ‰ ωpupbqq.
Proof. Let a ‰ b P N. There exists s P E with s ‰ 1 such that apsq ‰ bpsq. Write s as a concatenation e i1¨¨¨¨¨ein , where i 1 , . . . , i n P t1, 2u. For each j " 1, . . . , n, there is an associated unary term u j such that, for all w P E, pu ij paqqpwq " pa˝γ ij qpwq " apw¨e ij q.
Write u in˝. . .˝u i1 as u s . Then u s pcqp1q " cpsq for all c P N and hence pω˝u s qpaq " u s paqp1q " apsq ‰ bpsq " u s pbqp1q " pω˝u s qpbq.
5 , then there exists a unary term function t of N such that ωptpaqq " 1 and ωptpbqq " 0.
Theorem 5.6 (Strong Duality Theorem for Double Ockham Algebras). Let N " pt0, 1u E ; _,^, f, g, 0, 1q and N " " pt0, 1u E ; u 1 , u 2 , ď, Tq be as defined above. Let ω P DpN 5 , 2q X PpN " 5 , 2 " q be given by evaluation at 1, the identity of the monoid E. Let D : DO Ñ Y and E : Y Ñ DO be the hom-functors: D :" DOp´, Nq and E :" Yp´, N " q. Then N " strongly dualises N, that is, D and E establish a strong duality between DO and Y. Moreover DpAq 5 -HpA 5 q in P and EpYq 5 -KpY 5 q in D, for A P DO and Y P Y, where the isomorphisms are set up by Φ A ω : x Þ Ñ ω˝x, for x P DpAq, and Ψ Y ω : α Þ Ñ ω˝α, for α P EpYq. Proof. We simply need to confirm that the conditions of [12,Theorem 4.4] are satisfied. We have everything set up to ensure that all the functors work as the theorem requires. In addition Lemmas 5.3-5.5 tell us that Conditions (1)-(3) in the theorem are satisfied. Some remarks are in order here. We stress that it is critical that we could find a map ω which acts as a morphism both on the algebra side and on the dual side, and has the separation properties set out in Lemmas 5.4 and 5.5. We also observe that for our application of [12,Theorem 4.4], its Condition (3) is met in a simpler way than the theorem allows for: the special form of the f, g (viz. dual endomorphisms with respect to the bounded lattice operations) that forces pω, ωq´1pďq to contain just one maximal subalgebra.
We should comment too on how our natural duality for DO relates to a Priestleystyle duality for DO. The latter can be set up in just the same way as that for O originating in [28]. This duality is an enrichment of that between D and P, whereby f and g are captured on the dual side via a pair of order-reversing continuous maps p and q, and morphisms are required to preserve these maps. Theorem 5.6 tells us that, for any A P DO, there is an isomorphism between the Priestley space reduct DpAq 5 of the natural dual of A P DO and the Priestley dual HpA 5 q of the D-reduct of A. Both these Priestley spaces carry additional structure: u 1 and u 2 in the former case and p and q in the latter. When the reducts of the natural and Priestley-style dual spaces of the algebras are identified these pairs of maps coincide. Thus the two dualities for DO are essentially the same and one may toggle between them at will. We have a new example here of a 'best of both worlds' scenario, in which we have both the advantages of a natural duality and the benefits, pictorially, of a duality based on Priestley spaces. See [7,Section 3], [8,Section 6] and [12,Section 4] for earlier recognition of occurrences of this phenomenon: other varieties for which it arises are De Morgan algebras and Ockham algebras. In general it is not hereditary: it fails to occur for Kleene algebras, for example.
Combining our results we arrive at our duality for the variety DB´.
Theorem 5.7 (Strong Duality Theorem for Bounded Distributive Bilattices with Generalised Conflation). Let N " " pt0, 1u E ; u 1 , u 2 , ď, Tq be as in Theorem 5.6. Then N "ˆN " yields a strong duality on DB´. Moreover the dual category for this duality is Y :" IS c P`pN " q which may, in turn, be identified with the category P DO of double Ockham spaces.
To illustrate the rewards derived from a natural duality for F DB´, we highlight the simple description of free objects that follows from Theorem 5.7: for a nonempty set S, the free algebra F DB´p Sq on S has pN " 2 q S as its natural dual space. Hence F DB´p Sq can be identified with the family of continuous structure-preserving maps from pN " 2 q S into N " 2 , with the operations defined pointwise. (Recall the remark on free algebras in Section 3.)

Dualities for pre-bilattice-based varieties
In this final section we consider dualities for pre-bilattice-based varieties. Here we call on the adaptation of the product representation theorem given in [9, Theorem 9.1]. Hitherto in this paper we have worked with dualities for prevarieties of the form ISPpMq, thereby encompassing dualities for many classes of interest in the context of bilattices. However when we drop negation and so move from bilattices to pre-bilattices the situation changes and we encounter classes of the form ISPpMq, where M is a finite set of algebras over a common language. For example, for distributive pre-bilattices M consists of a pair of two-element algebras, one with truth and knowledge orders equal, the other with these as order duals. Fortunately a form of natural duality theory exists which is applicable to classes of the form ISPpMq; this makes use of multisorted structures on the dual side. So in this section we shall consider dualities for pre-bilattice-based varieties. As a starting point we have the treatment of distributive pre-bilattices given in [8, Sections 9 and 10]; a self-contained summary of the rudiments of multisorted duality theory can also be found there or see [10,Chapter 7].
We first recall how [9, Theorem 9.1] differs from Theorem 2.1. We start from a base class VpN q, where N is a class of algebras over a common language Σ. Let Γ and P Γ pN q be as in Section 2. Negation in a product bilattice links the two factors, and condition (P) from the definition of duplication by Γ reflects this. In the absence of negation, (P) is dropped and the following condition is substituted: (D) for pt 1 , t 2 q P Γ with n pt1,t2q " n, there exist n-ary Σ-terms r 1 and r 2 such that t 1 px 1 , . . . , x 2n q " r 1 px 1 , x 3 , . . . , x 2n´1 q and t 2 px 1 , . . . , x 2n q " r 2 px 2 , x 4 , . . . , x 2n q.
A product algebra associated with Γ now takes the form P d Γ Q " pPˆQ; trt 1 , t 2 s Pd Γ Q | pt 1 , t 2 q P Γuq, where P, Q belong to the base variety B " VpN q. This construction is used to define a functor d Γ : BˆB Ñ A as follows: on objects: pP, Qq Þ Ñ P d Γ Q, on morphisms: d Γ ph 1 , h 2 qpa, bq " ph 1 paq, h 2 pbqq. We move on to consider dualities for duplicated varieties. For simplicity we shall first assume that the base variety B " ISPpNq has a single-sorted duality with alter ego N " " pN ; G, R, Tq. Our next task is to determine a set of generators for A as a prevariety. We denote the trivial algebra by T. For C P B let fC : C Ñ T be the unique homomorphism from C into T. Lemma 6.2. If B " ISPpNq " VpNq for some algebra N, then Proof. Let A P A and a ‰ b P A. By Theorem 6.1, we may assume that there exist B, C P B such that A " B d Γ C. Let a 1 , b 1 P B and a 2 , b 2 P C such that a " pa 1 , a 2 q and b " pb 1 , b 2 q. By simmetry we may assume that a 1 ‰ b 1 . Then there exists a homomorphism h : Let M " tN d Γ T, T d Γ Nu. We now 'double up' N " in the obvious way. Let N " Z N " " pN 1 9 YN 2 ; G 1 , G 2 , R 1 , R 2 , Tq, based on disjointified universes N 1 and N 2 , such that pN i ; G i , R i , Tae Ni q is isomorphic to N " for i " 1, 2. Identify N 1 with NˆT and N 2 with TˆN and define M " " N " Z N " .
We now present our transfer theorem for natural dualities associated with Theorem 6.1 (the single-sorted case). Its proof is largely a diagram-chase with functors. Below, Id C denotes the identity functor on a category C andis used to denote natural isomorphism. Theorem 6.3. Let N be a Σ-algebra and assume that Γ satisfies (L), (M) and (D) relative to N. Assume that N " " pN ; G, R, Tq yields a duality on B " ISPpNq " VpNq with dual category Y " IS c P`pN " q. Let M and M " be defined as above. Then M " yields a multisorted duality for A " ISPpMq " VpP d Γ Q | P, Q P VpN qq for which the dual category is X -YˆY. If the duality for B is full, respectively strong, then the same is true of that for A.
Proof. Let pX 1 , X 2 q P YˆY " IS c P`pN " qˆIS c P`pN " q. We identify this structure with X 1 Z X 2 " pX 1 9 YX 2 ; G 1 , G 2 , R 1 , R 2 , Tq, where as before 9 Y denotes disjoint union and the topology T is the union of T 1 and T 2 . Morphisms in X are maps f : X 1 9 YX 2 Ñ Y 1 9 YY 2 that respect the structure and are such that f pxq P Y i when x P X i and i P t1, 2u. Hence the assignment: on objects: on morphisms: pf, gq Þ Ñ f 9 Yg sets up a categorical equivalence, Z. Let F : X Ñ YˆY denote its inverse. Identify N d Γ T and T d Γ N with N 1 and N 2 respectively. One sees that M " :" N " Z N " " pN 1 9 YN 2 ; G 1 , G 2 , R 1 , R 2 , Tq is a legitimate alter ego for M. Let D B : B Ñ Y and E B : Y Ñ B, and D A : A Ñ X and E A : X Ñ A be the homfunctors determined by N " and M " respectively. By Theorem 6.1, there exists a Figure 1. Natural duality by duplication functor C : A Ñ BˆB that together with d Γ : BˆB Ñ A determines a categorical equivalence. Take A, B P B and let D A pA d Γ Bq " pX 1 9 YX 2 ; G 1 , G 2 , R 1 , R 2 , Tq.
For an n-ary relation r P R, let r Ad Γ B i be the corresponding relation in R Ad Γ B i Ď X n i (i " t1, 2u). So ph 1 , . . . , h n q P r Ad Γ B 1 if and only if h i " pg i , fBq P BpA, Nqt fBu for i P t1, . . . , nu and pg 1 , . . . , g n q P r A . Similarly, a tuple ph 1 , . . . , h n q belongs to r Ad Γ B 2 if and only if h i " pfÅ, g i q P tfÅuˆBpB, Nq for i P t1, . . . , nu and pg 1 , . . . , g n q P r B . The same argument applied to G proves that pX 1 ; G 1 , R 1 , Tae X1 q and pX 2 ; G 2 , R 2 , Tae X2 q are isomorphic to D B pAq and D B pBq, respectively. Thus FpD A pA d Γ Bqq is isomorphic to pD B pAq, D B pBqq in YˆY. Moreover, it is easy to see that the assignment FpD A pA d Γ Bqq Þ Ñ pD B pAq, D B pBqq determines a natural isomorphism between F˝D A˝dΓ and D BˆDB : BˆB Ñ XˆX.
Similarly, for each pX, Yq P XˆX, Moreover, the assignment E A pX Z Yq Þ Ñ E B pXq d Γ E B pYq is natural in X and Y, that is, E A˝Z -pE BˆEB q˝d Γ . So (up to natural isomorphism) the diagrams in Figure 1 commute. A symbolchase now confirms that M " dualises M because N " dualises N: Figure 2. Full duality by duplication Assume that N " yields a full duality. Then the diagram in Figure 2 commutes. We can easily prove that D A˝EA -Id X , that is, M " yields a full duality. Moreover, if N " is injective in Y then pN " , N " q is injective in YˆY, or equivalently M " " N " Z N " is injective in X. Hence M " yields a strong duality if N does. Theorem 6.3 applies to the variety pDBu of (unbounded) distributive pre-bilattices. Its members are algebras A " pA; _ t ,^t, _ k ,^kq for which pA; _ t ,^tq P D u and pA; _ k ,^kq P D u . The well-known product representation for pDBu comes from the observation that the set Γ pDBu " tp_ 4 13 ,^4 24 q, p^4 13 , _ 4 24 q, p_ 4 13 , _ 4 24 q, p^4 13 ,^4 24 qu satisfies (L), (M) and (D) [9, Section 9]. Since 2 " u strongly dualises D u , the structure 2 " u Z 2 " u determines a multisorted strong duality for pDBu. This was established by different techniques in [8,Theorem 10.2]. Theorem 6.3 also yields dualities for distributive trilattices. These are (to the best of our knowledge) new. As with pre-bilattices, we opt for the unbounded case. An unbounded distributive trilattice is an algebra pA; _ t ,^t, _ f ,^f , _ i ,^iq such that pA; _ t ,^tq, pA; _ f ,^f q and pA; _ i ,^iq are distributive lattices. Let DT u denote the variety of (unbounded) distributive trilattices. An algebra pA; _ t ,^t, _ f ,^f , _ i ,^i,´tq is a distributive trilattice with t-involution if pA; _ t ,^t, _ f ,^f , _ i ,^iq P DT u and Γ DT´t " tpp^tq 4 13 , p^tq 4 24 q, pp_ t q 4 13 , p_ t q 4 24 q, pp_ k q 4 13 , p^kq 4 24 q, pp^kq 4 13 , p_ k q 4 24 q, pp^kq 4 13 , p^kq 4 24 q, pp_ k q 4 13 , p_ k q 4 24 q, p 2 1 , 2 2 qu. Then Γ DT´t satisfies (L), (M) and (D) over DBu (see [8,Example 9.4]). In Section 4, we used Theorem 3.1 to prove that p2 " u q 2 yields a strong duality on DBu. Now Theorem 6.3 implies that p2 " u q 2 Z p2 " u q 2 determines a multisorted strong duality for unbounded distributive trilattices with t-involution.
We can easily adapt our results to cater for a base variety which admits a multisorted duality rather than a single-sorted one. Predictably this leads to multisortedness at the duplicate level. In the case of Theorem 3.1, one obtains the required alter ego by squaring the base level alter ego, sort by sort; as before, the base variety and its duplicate have the same dual category. The extension of Theorem 6.3 employs two disjoint copies of each sort of the base-level alter ego. The proofs of these results involve only minor modifications of those for the single-sorted case. As an example, the multisorted version of Theorem 6.3 combined with the results in [9, Example 9.4] leads to a strong duality for unbounded distributive trilattices which has four sorts, obtained from the two-sorted duality for pDBu.