Priestley Duality Research Articles

Algebraic frames in Priestley duality

Algebraic frames in Priestley duality

On duality and model theory for polyadic spaces

On duality and model theory for polyadic spaces

Deriving Dualities in Pointfree Topology from Priestley Duality

Deriving Dualities in Pointfree Topology from Priestley Duality

Duality theory for enriched Priestley spaces

Duality theory for enriched Priestley spaces

Remarks on hyperspaces for Priestley spaces

Remarks on hyperspaces for Priestley spaces

Hofmann–Mislove through the lenses of Priestley

We use Priestley duality to give a new proof of the Hofmann–Mislove Theorem.

The exact completion for regular categories enriched in posets

The exact completion for regular categories enriched in posets

Expanding Belnap 2: the dual category in depth

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’(⊤) and ‘no information’ (⊥), 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, Jn, for n ⩾ 0, with J0 being Belnap’s seminal example. We gave a duality for the variety Vn generated by Jn, with the dual category Xn consisting of multi-sorted topological structures. Here we study the dual category in depth. We axiomatise the category Xn and show that it is isomorphic to a category Yn of single-sorted topological structures. The objects of Yn are ranked Priestley spaces endowed with a continuous retraction. We show how to construct the Priestley dual of the underlying bounded distributive lattice of an algebra in Vn via its dual in Yn; as an application we show that the size of the free algebra FVn(1) is given by a polynomial in n of degree 6.

Spectral properties of cBCK-algebras

In this paper we study prime spectra of commutative BCK-algebras. We give a new construction for commutative BCK-algebras using rooted trees, and determine both the ideal lattice and prime ideal lattice of such algebras. We prove that the spectrum of any commutative BCK-algebra is a locally compact generalized spectral space which is compact if and only if the algebra is finitely generated as an ideal. Further, we show that if a commutative BCK-algebra is involutory, then its spectrum is a Priestley space. Finally, we consider the functorial properties of the spectrum and define a functor from the category of commutative BCK-algebras to the category of distributive lattices with zero. We give a partial answer to the question: what distributive lattices lie in the image of this functor?

Vietoris hyperspaces over scattered Priestley spaces

We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{Skula} if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply our results to scattered compact spaces based on certain almost disjoint families, in particular, Lusin families and ladder systems.

Stone dualities from opfibrations

Stone dualities from opfibrations

Natural and restricted Priestley duality for ternary algebras and their cousins

Natural and restricted Priestley duality for ternary algebras and their cousins

Lattice Structures for Attractors III

The theory of bounded, distributive lattices provides the appropriate language for describing directionality and asymptotics in dynamical systems. For bounded, distributive lattices the general notion of ‘set-difference’ taking values in a semilattice is introduced, and is called the Conley form. The Conley form is used to build concrete, set-theoretic models of spectral spaces, or Priestley spaces, of bounded, distributive lattices and their finite coarsenings. Such representations formulate and compute order-theoretic models of dynamical systems such as Morse decompositions and Morse representations, which may be regarded as global characteristics of a dynamical system.

Priestley duality for MV-algebras and beyond

Abstract 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.

An isomorphic approach of fuzzy soft lattices to fuzzy soft Priestley spaces

The main purpose of this paper is to establish a relation between fuzzy soft lattices and fuzzy soft Priestley spaces. We have proved that each bounded distributive lattice of fuzzy soft sets is isomorphic to the lattice of all clopen upsets of a Priestley space. For this reason, we have defined fuzzy soft upsets, fuzzy soft downsets, fuzzy soft filters, fuzzy soft ideals and fuzzy soft Priestley space. To endorse the above relation, we have proved some related results.

Distributive lattices with strong endomorphism kernel property as direct sums

Unbounded distributive lattices which have strong endomorphism kernel property (SEKP) introduced by Blyth and Silva in [3] were fully characterized in [11] using Priestley duality (see Theorem 2.8}). We shall determine the structure of special elements (which are introduced after Theorem 2.8 under the name strong elements) and show that these lattices can be considered as a direct product of three lattices, a lattice with exactly one strong element, a lattice which is a direct sum of 2 element lattices with distinguished elements 1 and a lattice which is a direct sum of 2 element lattices with distinguished elements 0, and the sublattice of strong elements is isomorphic to a product of last two mentioned lattices.

Varieties of Regular Pseudocomplemented de Morgan Algebras

In this paper, we investigate the varieties $\mathbf M_n$ and $\mathbf K_n$ of regular pseudocomplemented de Morgan and Kleene algebras of range $n$, respectively. Priestley duality as it applies to pseudocomplemented de Morgan algebras is used. We characterise the dual spaces of the simple (equivalently, subdirectly irreducible) algebras in $\mathbf M_n$ and explicitly describe the dual spaces of the simple algebras in $\mathbf M_1$ and $\mathbf K_1$. We show that the variety $\mathbf M_1$ is locally finite, but this property does not extend to $\mathbf M_n$ or even $\mathbf K_n$ for $n \geq 2$. We also show that the lattice of subvarieties of $\mathbf K_1$ is an $\omega + 1$ chain and the cardinality of the lattice of subvarieties of either $\mathbf K_2$ or $\mathbf M_1$ is $2^{\omega}$. A description of the lattice of subvarieties of $\mathbf M_1$ is given.

Difference hierarchies and duality with an application to formal languages

Difference hierarchies and duality with an application to formal languages

Characterization of metrizable Esakia spaces via some forbidden configurations

By Priestley duality, each bounded distributive lattice is represented as the lattice of clopen upsets of a Priestley space, and by Esakia duality, each Heyting algebra is represented as the lattice of clopen upsets of an Esakia space. Esakia spaces are those Priestley spaces that satisfy the additional condition that the downset of each clopen is clopen. We show that in the metrizable case Esakia spaces can be singled out by forbidding three simple configurations. Since metrizability yields that the corresponding lattice of clopen upsets is countable, this provides a characterization of countable Heyting algebras. We show that this characterization no longer holds in the uncountable case. Our results have analogues for co-Heyting algebras and bi-Heyting algebras, and they easily generalize to the setting of p-algebras.

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.