Theory Of Natural Dualities Research Articles

Perfect extensions of de Morgan algebras

An algebra \(\mathbf{A}\) is called a perfect extension of its subalgebra \(\mathbf{B}\) if every congruence of \(\mathbf{B}\) has a unique extension to \(\mathbf{A}\). This terminology was used by Blyth and Varlet [1994]. In the case of lattices, this concept was described by Grätzer and Wehrung [1999] by saying that \(\mathbf{A}\) is a congruence-preserving extension of \(\mathbf{B}\). Not many investigations of this concept have been carried out so far. The present authors in another recent study faced the question of when a de Morgan algebra \(\mathbf{M}\) is perfect extension of its Boolean subalgebra \(B(\mathbf{M})\), the so-called skeleton of \(\mathbf{M}\). In this note a full solution to this interesting problem is given. The theory of natural dualities in the sense of Davey and Werner [1983] and Clark and Davey [1998], as well as Boolean product representations, are used as the main tools to obtain the solution.

Bitopological duality for algebras of Fitting’s logic and natural duality extension

In this paper, we investigate a bitopological duality for algebras of Fitting’s multi-valued logic. We also extend the natural duality theory for $$\mathbb {ISP_I}(\mathcal {L})$$ by developing a duality for $$\mathbb {ISP}(\mathcal {L})$$ , where $$\mathcal {L}$$ is a finite algebra in which underlying lattice is bounded distributive.

Sugihara algebras and Sugihara monoids: Multisorted dualities

AbstractThe authors developed in a recent paper natural dualities for finitely generated quasivarieties of Sugihara algebras. They thereby identified the admissibility algebras for these quasivarieties which, via the Test Spaces Method devised by Cabrer et al., give access to a viable method for studying admissible rules within relevance logic, specifically for extensions of the deductive system R-mingle.This paper builds on the work already done on the theory of natural dualities for Sugihara algebras. Its purpose is to provide an integrated suite of multisorted duality theorems of a uniform type, encompassing finitely generated quasivarieties and varieties of both Sugihara algebras and Sugihara monoids, and embracing both the odd and the even cases. The overarching theoretical framework of multisorted duality theory developed here leads on to amenable representations of free algebras. More widely, it provides a springboard to further applications.

Sugihara algebras: Admissibility algebras via the test spaces method

This paper studies finitely generated quasivarieties of Sugihara algebras. These quasivarieties provide complete algebraic semantics for certain propositional logics associated with the relevant logic R-mingle. The motivation for the paper comes from the study of admissible rules. Recent earlier work by the present authors, jointly with Freisberg and Metcalfe, laid the theoretical foundations for a feasible approach to this problem for a range of logics—the Test Spaces Method. The method, based on natural duality theory, provides an algorithm to obtain the algebra of minimum size on which admissibility of sets of rules can be tested. (In the most general case a set of such algebras may be needed rather than just one.) The method enables us to identify this ‘admissibility algebra’ for each quasivariety of Sugihara algebras which is generated by an algebra whose underlying lattice is a finite chain. To achieve our goals, it was first necessary to develop a (strong) duality for each of these quasivarieties. The dualities promise also to provide a valuable new tool for studying the structure of Sugihara algebras more widely.

Dualisability of partial unars

The dualisability of partial algebras is a largely unexplored area within natural duality theory. This paper considers the dualisability of finite structures that have a single partial unary operation in the type. We show that every such finite partial unar is dualisable. We obtain this result by showing that the relational structure obtained by replacing the fundamental operation by its graph is dualisable. We also give a finite generator for the class of all disjoint unions of directed trees up to some fixed height, considered as partial unars.

New-from-old full dualities via axiomatisation

We study different full dualities based on the same finite algebra. Our main theorem gives conditions on two different alter egos of a finite algebra under which, if one yields a full duality, then the other does too. We use this theorem to obtain a better understanding of several important examples from the theory of natural dualities. We also clarify what it means for two full dualities based on the same finite algebra to be different. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual categories.

MV-algebras, infinite dimensional polyhedra, and natural dualities

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.

Piggyback dualities revisited

In natural duality theory, the piggybacking technique is a valuable tool for constructing dualities. As originally devised by Davey and Werner, and extended by Davey and Priestley, it can be applied to finitely generated quasivarieties of algebras having term-reducts in a quasivariety for which a well-behaved natural duality is already available. This paper presents a comprehensive study of the method in a much wider setting: piggyback duality theorems are obtained for suitable prevarieties of structures. For the first time, and within this extended framework, piggybacking is used to derive theorems giving criteria for establishing strong dualities and two-forone dualities. The general theorems specialise in particular to the familiar situation in which we piggyback on Priestley duality for distributive lattices or Hofmann–Mislove– Stralka duality for semilattices, and many well-known dualities are thereby subsumed. A selection of new dualities is also presented.

Bohr Compactifications of Algebras and Structures

This paper provides a unifying framework for a range of categorical constructions characterised by universal mapping properties, within the realm of compactifications of discrete structures. Some classic examples fit within this broad picture: the Bohr compactification of an abelian group via Pontryagin duality, the zero-dimensional Bohr compactification of a semilattice, and the Nachbin order-compactification of an ordered set. The notion of a natural extension functor is extended to suitable categories of structures and such a functor is shown to yield a reflection into an associated category of topological structures. Our principal results address reconciliation of the natural extension with the Bohr compactification or its zero-dimensional variant. In certain cases the natural extension functor and a Bohr compactification functor are the same; in others the functors have different codomains but may agree on all objects. Coincidence in the stronger sense occurs in the zero-dimensional setting precisely when the domain is a category of structures whose associated topological prevariety is standard. It occurs, in the weaker sense only, for the class of ordered sets and, as we show, also for infinitely many classes of ordered structures. Coincidence results aid understanding of Bohr-type compactifications, which are defined abstractly. Ideas from natural duality theory lead to an explicit description of the natural extension which is particularly amenable for any prevariety of algebras with a finite, dualisable, generator. Examples of such classes—often varieties—are plentiful and varied, and in many cases the associated topological prevariety is standard.

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).

Naturally dualizable algebras omitting types 1 and 5 have a cube term

An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if A omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then A has a cube term.

Similarity and a Duality for Fullerenes

Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality.

Automorphism-primal algebras generate verbose varieties

A finite algebra is called automorphism-primal if its clone of term operations coincides with all operations that preserve its automorphisms. We prove that the variety generated by an automorphism-primal algebra is verbose, that is, on every member algebra, every fully invariant congruence is verbal. The proof is a nice application of the theory of natural dualities as developed by Davey et al.

Distributive bilattices from the perspective of natural duality theory

This paper provides a fresh perspective on the representation of distributive bilattices and of related varieties. The techniques of natural duality are employed to give, economically and in a uniform way, categories of structures dually equivalent to these varieties. We relate our dualities to the product representations for bilattices and to pre-existing dual representations by a simple translation process which is an instance of a more general mechanism for connecting dualities based on Priestley duality to natural dualities. Our approach gives us access to descriptions of algebraic/categorical properties of bilattices and also reveals how ‘truth’ and ‘knowledge’ may be seen as dual notions.

Variable-basis Categorically-algebraic Dualities

The manuscript continues our study on developing a categorically-algebraic (catalg) analogue of the theory of natural dualities of D. Clark and B. Davey, which provides a machinery for obtaining topological representations of algebraic structures. The new setting differs from its predecessor in relying on catalg topology, introduced lately by the author as a new approach to topological structures, which incorporates the majority of both crisp and many-valued developments, ultimately erasing the border between them. Motivated by the variable-basis lattice-valued extension of the Stone representation theorems done by S. E. Rodabaugh, we have recently presented a catalg version of the Priestley duality for distributive lattices, which gave rise (as in the classical case) to a fixed-basis variety-based approach to natural dualities. In this paper, we extend the theory to variable-basis, whose setting is completely different from the respective one of S. E. Rodabaugh, restricted to isomorphisms between the underlying lattices of the spaces.

Product representation for default bilattices: An application of natural duality theory

Bilattices (that is, sets with two lattice structures) provide an algebraic tool to model simultaneously the validity of, and knowledge about, sentences in an appropriate language. In particular, certain bilattices have been used to model situations in which information is prioritised and so can be viewed hierarchically. These default bilattices are not interlaced: the lattice operations of one lattice structure do not preserve the order of the other one. The well-known product representation theorem for interlaced bilattices does not extend to bilattices which fail to be interlaced and the lack of a product representation has been a handicap to understanding the structure of default bilattices. In this paper we study, from an algebraic perspective, a hierarchy of varieties of default bilattices, allowing for different levels of default. We develop natural dualities for these varieties and thereby obtain a concrete representation for the algebras in each variety. This leads on to a form of product representation that generalises the product representation as this applies to distributive bilattices.

Coproducts of distributive lattice-based algebras

This paper presents a systematic study of coproducts. This is carried out principally, but not exclusively, for finitely generated quasivarieties $${{\bf \mathcal{A}}}$$ that admit a (term) reduct in the variety $${{\bf \mathfrak{D}}}$$ of bounded distributive lattices. In this setting, we present necessary and sufficient conditions on $${{\bf \mathcal{A}}}$$ for the forgetful functor $${U_{{\bf \mathcal{A}}}}$$ from $${{\bf \mathcal{A}}}$$ to $${{\bf \mathfrak{D}}}$$ to preserve coproducts. We also investigate the possible behaviours of $${U_{{\bf \mathcal{A}}}}$$ as regards coproducts in $${{\bf \mathcal{A}}}$$ under weaker assumptions. Depending on the properties exhibited by the functor, different procedures are then available for describing these coproducts. We classify a selection of well–known varieties within our scheme, thereby unifying earlier results and obtaining some new ones. The paper’s methodology draws heavily on duality theory. We use Priestley duality as a tool and our descriptions of coproducts are given in terms of this duality. We also exploit natural duality theory, specifically multisorted piggyback dualities, in our analysis of the behaviour of the forgetful functor into $${{\bf \mathfrak{D}}}$$ . In the opposite direction, we reveal that the type of natural duality that the class $${{\bf \mathcal{A}}}$$ can possess is governed by properties of coproducts in $${{\bf \mathcal{A}}}$$ and the way in which the classes $${{\bf \mathcal{A}}}$$ and $${U_{{\bf \mathcal{A}}}}$$ ( $${{\bf \mathcal{A}}}$$ ) interact.

A Fresh Perspective on Canonical Extensions for Bounded Lattices

This paper presents a novel treatment of the canonical extension of a bounded lattice, in the spirit of the theory of natural dualities. At the level of objects, this can be achieved by exploiting the topological representation due to M. Ploščica, and the canonical extension can be obtained in the same manner as can be done in the distributive case by exploiting Priestley duality. To encompass both objects and morphisms the Ploščica representation is replaced by a duality due to Allwein and Hartonas, recast in the style of Ploščica’s paper. This leads to a construction of canonical extension valid for all bounded lattices, which is shown to be functorial, with the property that the canonical extension functor decomposes as the composite of two functors, each of which acts on morphisms by composition, in the manner of hom-functors.

THE LATTICE OF ALTER EGOS

We introduce a new Galois connection for partial operations on a finite set, which induces a natural quasi-order on the collection of all partial algebras on this set. The quasi-order is compatible with the basic concepts of natural duality theory, and we use it to turn the set of all alter egos of a given finite algebra into a doubly algebraic lattice. The Galois connection provides a framework for us to develop further the theory of natural dualities for partial algebras. The development unifies several fundamental concepts from duality theory and reveals a new understanding of full dualities, particularly at the finite level.

Natural duality, modality, and coalgebra

The theory of natural dualities is a general theory of Stone–Priestley-type categorical dualities based on the machinery of universal algebra. Such dualities play a fundamental role in recent developments of coalgebraic logic. At the same time, however, natural duality theory has not subsumed important dualities in coalgebraic logic, including the Jónsson–Tarski topological duality and the Abramsky–Kupke–Kurz–Venema coalgebraic duality for the class of all modal algebras. By introducing a new notion of I S P M , in this paper, we aim to extend the theory of natural dualities so that it encompasses the Jónsson–Tarski duality and the Abramsky–Kupke–Kurz–Venema duality. The main results are topological and coalgebraic dualities for I S P M ( L ) where L is a semi-primal algebra with a bounded lattice reduct. These dualities are shown building upon the Keimel–Werner semi-primal duality theorem. Our general theory subsumes both the Jónsson–Tarski and Abramsky–Kupke–Kurz–Venema dualities. Moreover, it provides new coalgebraic dualities for algebras of many-valued modal logics and certain insights into a category-equivalence problem for categories of algebras involved. It also follows from our dualities that the corresponding categories of coalgebras have final coalgebras and cofree coalgebras. I S P M provides a natural framework for the universal algebra of modalities, and as such, for the theory of modal natural dualities.