Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

A Duality Theoretic View on Limits of Finite Structures

A systematic theory of structural limits for finite models has been developed by Nešetřil and Ossona de Mendez. It is based on the insight that the collection of finite structures can be embedded, via a map they call the Stone pairing, in a space of measures, where the desired limits can be computed. We show that a closely related but finer grained space of measures arises — via Stone-Priestley duality and the notion of types from model theory — by enriching the expressive power of first-order logic with certain “probabilistic operators”. We provide a sound and complete calculus for this extended logic and expose the functorial nature of this construction.The consequences are two-fold. On the one hand, we identify the logical gist of the theory of structural limits. On the other hand, our construction shows that the duality-theoretic variant of the Stone pairing captures the adding of a layer of quantifiers, thus making a strong link to recent work on semiring quantifiers in logic on words. In the process, we identify the model theoretic notion of types as the unifying concept behind this link. These results contribute to bridging the strands of logic in computer science which focus on semantics and on more algorithmic and complexity related areas, respectively.

On Endomorphism Monoids of Finite Kleene Algebras

An endomorphism monoid of an algebra [Formula: see text] is said to be a band if every endomorphism on [Formula: see text] is an idempotent, and it is said to be a demi-band if every non-injective endomorphism on [Formula: see text] is an idempotent. We precisely determine finite Kleene algebras whose endomorphism monoids are demi-bands and bands via Priestley duality.

Regular double p-algebras

In this paper, we investigate the variety RDP of regular double p-algebras and its subvarieties RDPn, n ≥ 1, of range n. First, we present an explicit description of the subdirectly irreducible algebras (which coincide with the simple algebras) in the variety RDP1 and show that this variety is locally finite. We also show that the lattice of subvarieties of RDP1, LV(RDP1), is isomorphic to the lattice of down sets of the poset {1} ⊕ (ℕ × ℕ). We describe all the subvarieties of RDP1 and conclude that LV(RDP1) is countably infinite. An equational basis for each proper subvariety of RDP1 is given. To study the subvarieties RDPn with n ≥ 2, Priestley duality as it applies to regular double p-algebras is used. We show that each of these subvarieties is not locally finite. In fact, we prove that its 1-generated free algebra is infinite and that the lattice of its subvarieties has cardinality 2ℵ0. We also use Priestley duality to prove that RDP and each of its subvarieties RDPn are generated by their finite members.

Spectral-like duality for Distributive Hilbert Algebras with Infimum

We present the results of our research on stone-type dualities for certain classes of ordered algebras that do not fall within the scope of extended Priestley-duality. In a forthcoming paper we study a new spectral-like duality for the class of distributive Hilbert algebras with infimum. We explain the main facts of that duality and we outline how the same strategy could be used for getting a Priestley-style duality for the same class of algebras, as well as dualities for other classes of algebras.

Priestley duality for (modal) N4-lattices

N4-lattices are the algebraic semantics of paraconsistent Nelson logic, which was introduced as an inconsistency-tolerant counterpart of the better-known logic of Nelson. Paraconsistent Nelson logic combines interesting features of intuitionistic, classical and many-valued logics (e.g., Belnap-Dunn four-valued logic); recent work has shown that it can also be seen as one member of the wide family of substructural logics.The work we present here is a contribution towards a better topological understanding of the algebraic counterpart of paraconsistent Nelson logic, namely a variety of involutive lattices called N4-lattices.

Compact Intersection Property and description of congruence lattices

We say that a variety V of algebras has the Compact Intersection Property (CIP), if the family of compact congruences of every A ∈ V is closed under intersection. We investigate the congruence lattices of algebras in locally finite congruence-distributive CIP varieties. We prove some general results and obtain a complete characterization for some types of such varieties. We provide two kinds of description of congruence lattices: via direct limits and via Priestley duality.

The Permutable Congruences of bpO-Algebras

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several classes of algebras, for example, de Morgan algebras, p-algebras and MS-algebras, etc. In this paper, we characterize the permutable congruences on a finite Ockham algebra with balanced pseudocomplementation via the Priestley duality.