Motivated by Haviar and Ploščica’s 2021 characterisation of Boolean products of simple De Morgan algebras, we investigate Boolean products of simple algebras in filtral varieties. We provide two main theorems. The first yields Werner’s Boolean-product representation of algebras in a discriminator variety as an immediate application. The second, which applies to algebras in which the top congruence is compact, yields a generalisation of the Haviar–Ploščica result to semisimple varieties of Ockham algebras. The property of having factor principal congruences is fundamental to both theorems. While major parts of our general theorems can be derived from results in the literature, we offer new, self-contained and essentially elementary proofs.

Up to term equivalence, there are three ways to assign a nonemptyset C of constants to the three-element Kleene lattice, leading toternary algebras (C = {0, d, 1}), Kleene algebras (C = {0, 1}), and don’tknow algebras (C = {d}). Our focus is on ternary algebras. We derivea strong, optimal natural duality and the restricted Priestley duality forternary algebras and give axiomatisations of the dual categories. We applythese dualities in tandem to give straightforward and transparent proofsof some known results for ternary algebras. We also discuss, and in somecases prove, the corresponding dualities for Kleene lattices, Kleene algebrasand don’t know algebras.

Bilattices, which provide an algebraic tool for simultaneously modelling knowledge and truth, were introduced by N.D. Belnap in a 1977 paper entitled 'How a computer should think'. Prioritised default bilattices include not only Belnap's four values, for `true' ($t$), `false'($f$), `contradiction' ($\top$) and `no information' ($\bot$), but also indexed families of default values for simultaneously modelling degrees of knowledge and truth. Prioritised default bilattices have applications in a number of areas including artificial intelligence. In our companion paper, we introduced a new family of prioritised default bilattices, $\mathbf J_n$, for $n \in \omega$, with $\mathbf J_0$ being Belnap's seminal example. We gave a duality for the variety $\mathcal V_n$ generated by $\mathbf J_n$, with the objects of the dual category $\mathcal X_n$ being multi-sorted topological structures. Here we study the dual category in depth. We give an axiomatisation of the category $\mathcal X_n$ and show that it is isomorphic to a category $\mathcal Y_n$ of single-sorted topological structures. The objects of $\mathcal Y_n$ are Priestley spaces endowed with a continuous retraction in which the order has a natural ranking. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in $\mathcal V_n$ via its dual in $\mathcal Y_n$; as an application we show that the size of the free algebra $\mathbf F_{\mathcal V_n}(1)$ is given by a polynomial in $n$ of degree $6$.

We study splittings or lack of them, in lattices of subvarieties of some logic-related varieties. We present a general lemma, the non-splitting lemma, which when combined with some variety-specific constructions, yields each of our negative results: the variety of commutative integral residuated lattices contains no splitting algebras, and in the varieties of double Heyting algebras, dually pseudocomplemented Heyting algebras and regular double [Formula: see text]-algebras the only splitting algebras are the two-element and three-element chains.

Bilattices provide an algebraic tool with which to model simultaneously knowledge and truth. They were introduced by Belnap in 1977 in a paper entitled \emph{How a computer should think}. Belnap argued that instead of using a logic with two values, for `true' ($t$) and `false' ($f$), a computer should use a logic with two further values, for `contradiction' ($\top$) and `no information' ($\bot$). The resulting structure is equipped with two lattice orders, a \emph{knowledge order} and a \emph{truth order}, and hence is called a \emph{bilattice}. Prioritised default bilattices include not only values for `true' ($t_0$), `false' ($f_0$), `contradiction' and `no information', but also indexed families of default values, $t_1, \dots, t_n$ and $f_1, \dots, f_n$, for simultaneous modelling of degrees of knowledge and truth. We focus on a new family of prioritised default bilattices: $\mathbf J_n$, for $n \in \omega$. The bilattice $\mathbf J_0$ is precisely Belnap's seminal example. We address mathematical rather than logical aspects of our prioritised default bilattices. We obtain a single-sorted topological representation for the bilattices in the quasivariety $\mathcal J_n$ generated by $\mathbf J_n$, and separately a multi-sorted topological representation for the bilattices in the variety $\mathcal V_n$ generated by $\mathbf J_n$. Our results provide an interesting example where the multi-sorted duality for the variety has a simpler structure than the single-sorted duality for the quasivariety.

Each finite algebra A induces a lattice L A via the quasi-order → on the finite members of the variety generated by A , where B → C if there exists a homomorphism from B to C . In this paper, we introduce the question: ‘Which lattices arise as the homomorphism lattice L A induced by a finite algebra A ?’ Our main result is that each finite distributive lattice arises as L Q , for some quasi-primal algebra Q . We also obtain representations of some other classes of lattices as homomorphism lattices, including all finite partition lattices, all finite subspace lattices and all lattices of the form L ⊕ 1 , where L is an interval in the subgroup lattice of a finite group.

We clarify what it means for two full dualities based on the same algebra to be different. 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. Throughout the paper, a fundamental role is played by the universal Horn theory of the dual classes.

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.

Anyone who has ever worked with a variety~$\boldsymbol{\mathscr{A}}$ of algebras with a reduct in the variety of bounded distributive lattices will know a restricted Priestley duality when they meet one---but until now there has been no abstract definition. Here we provide one. After deriving some basic properties of a restricted Priestley dual category $\boldsymbol{\mathscr{X}}$ of such a variety, we give a characterisation, in terms of $\boldsymbol{\mathscr{X}}$, of finitely generated discriminator subvarieties of~$\boldsymbol{\mathscr{A}}$. As a first application of our characterisation, we give a new proof of Sankappanavar's characterisation of finitely generated discriminator varieties of distributive double p-algebras. A substantial portion of the paper is devoted to the application of our results to Cornish algebras. A Cornish algebra is a bounded distributive lattice equipped with a family of unary operations each of which is either an endomorphism or a dual endomorphism of the bounded lattice. They are a natural generalisation of Ockham algebras, which have been extensively studied. We give an external necessary-and-sufficient condition and an easily applied, completely internal, sufficient condition for a finite set of finite Cornish algebras to share a common ternary discriminator term and so generate a discriminator variety. Our results give a characterisation of discriminator varieties of Ockham algebras as a special case, thereby yielding Davey, Nguyen and Pitkethly's characterisation of quasi-primal Ockham algebras.

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.

We find all finite Ockham algebras that admit only finitely many compatible relations (modulo a natural equivalence). Up to isomorphism and symmetry, these Ockham algebras form two countably infinite families: one family consists of the quasi-primal Ockham algebras, and the other family is a sequence of generalised Stone algebras.

We make a start on one of George McNulty's Dozen Easy Problems: "Which finite automatic algebras are dualizable?" We give some necessary and some sufficient conditions for dualizability. For example, we prove that a finite automatic algebra is dualizable if its letters act as an abelian group of permutations on its states. To illustrate the potential difficulty of the general problem, we exhibit an infinite ascending chain $\mathbf A_1 \le \mathbf A_2 \le \mathbf A_3 \le ...b$ of finite automatic algebras that are alternately dualizable and non-dualizable.

This note aims to unravel the history of the Product Representation Theorem for Interlaced Pre-bilattices. We will see that it has its lattice-theoretic roots in early attempts to solve one of the problems in Birkhoff’s Lattice Theory. The theorem was presented in its full generality by Czédli, Huhn and Szabó at a conference in Szeged, Hungary in 1980 (and published in 1983). This was several years before Ginsberg introduced bilattices at a conference on artificial intelligence in 1986 and in his foundational paper in 1988.

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.

The paper investigates completions in the context of finitely generated lattice-based varieties of algebras. In particular the structure of canonical extensions in such a variety \({\mathcal {A}}\) is explored, and the role of the natural extension in providing a realisation of the canonical extension is discussed. The completions considered are Boolean topological algebras with respect to the interval topology, and consequences of this feature for their structure are revealed. In addition, we call on recent results from duality theory to show that topological and discrete dualities for \({\mathcal {A}}\) exist in partnership.

Traditionally in natural duality theory the algebras carry no topology and the objects on the dual side are structured Boolean spaces. Given a duality, one may ask when the topology can be swapped to the other side to yield a partner duality (or, better, a dual equivalence) between a category of topological algebras and a category of structures. A prototype for this procedure is provided by the passage from Priestley duality for bounded distributive lattices to Banaschewski duality for ordered sets. Moreover, the partnership between these two dualities yields as a spin-off a factorisation of the functor sending a bounded distributive lattice to its natural extension, alias , in this case, the canonical extension or profinite completion. The main theorem of this paper validates topology swapping as a uniform way to create new dual adjunctions and dual equivalences: we prove that, for every finite algebra of finite type, each dualising alter ego gives rise to a partner duality. We illustrate the theorem via a variety of natural dualities, some classic and some less familiar. For lattice-based algebras this leads immediately, as in the Priestley–Banaschewski example, to a concrete description of canonical extensions.

Given a full duality based on a finite algebra M, we show how to create a full duality based on any other finite algebra N for which \({\mathbb{ISP}({\bf M}) = \mathbb{ISP}({\bf N})}\). So the full dualisability of a quasivariety is independent of the algebra chosen as the generator. We obtain this result by proving the corresponding result for multisorted full dualities.

An algebra A has finite degree if its term functions are determined by some finite set of finitary relations on A. We study this concept for finite algebras in general and for finite semigroups in particular. For example, we show that every finite nilpotent semigroup has finite degree (more generally, every finite algebra with bounded p (n) -sequence), and every finite commutative semigroup has finite degree. We give an example of a five-element unary semigroup that has infinite degree. We also give examples to show that finite degree is not preserved in general under taking subalgebras, homomorphic images, direct products or subdirect factors.

This paper investigates profinite completions of residually finite algebras, drawing on ideas from the theory of natural dualities. Given a class \({\mathcal{A} = \mathbb{ISP}(\mathcal{M})}\), where \({\mathcal{M}}\) is a set, not necessarily finite, of finite algebras, it is shown that each \({{\bf A} \in \mathcal{A}}\) embeds as a topologically dense subalgebra of a topological algebra \({n_{\mathcal{A}}({\bf A})}\) (its natural extension), and that \({n_{\mathcal{A}}({\bf A})}\) is isomorphic, topologically and algebraically, to the profinite completion of A. In addition it is shown how the natural extension may be concretely described as a certain family of relation-preserving maps; in the special case that \({\mathcal{M}}\) is finite and \({\mathcal{A}}\) possesses a single-sorted or multisorted natural duality, the relations to be preserved can be taken to be those belonging to a dualising set. For an algebra belonging to a finitely generated variety of lattice-based algebras, it is known that the profinite completion coincides with the canonical extension. In this situation the natural extension provides a new concrete realisation of the canonical extension, generalising the well-known representation of the canonical extension of a bounded distributive lattice as the lattice of up-sets of the underlying ordered set of its Priestley dual. The paper concludes with a survey of classes of algebras to which the main theorems do, and do not, apply.

We prove that if a quasivariety \({\mathcal{A}}\) generated by a finite family \({\mathcal{M}}\) of finite algebras has a multisorted duality based on \({\mathcal{M}}\), then \({\mathcal{A}}\) has a multisorted duality based on any finite family of finite algebras that generates it.