Pseudocomplemented Distributive Lattices Research Articles

A 1971 question of Grätzer and Lakser on pseudocomplemented lattices

Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 2n ⊕ 1 can be characterized by the property of not having a * homomorphism onto 2i ⊕ 1 for 1 < i < n.In this article, this question is answered.

Median filters of pseudocomplemented distributive lattices

Median filters of pseudocomplemented distributive lattices

Residuated lattices derived from filters(ideals) in double Boolean algebras

Double Boolean algebras (dBas) are algebraic structures D = (D, v, ^, ‌, ', ⊥, T) of type (2, 2, 1, 1, 0, 0), introduced by Rudolf Wille to capture the equational theory of the algebra of protoconcepts. Our goal is an algebraic investigation of dBas, based on similar results on Boolean algebras. In this paper, first we characterize filters on dBas as deductive systems and we give many characterization of primary filters(ideals). Second, for a given dBa, we show that the set of its filters F(D) (resp.ideals I(D)) is endowed with the structure of distributive pseudo-complemented lattices, Heyting algebras and residuated lattices. We finish by introducing the notions of annihilators and co-annihilators on dBas and investigate some relalted properties of them. We show that pseudo-complement of an ideal I (filter F) is the annihilator I* of I ( co-annihilator F*) and the set of annihilators (co-annihilators) forms a Boolean algebra.

Logics of involutive Stone algebras

An involutive Stone algebra (IS-algebra) is simultaneously a De Morgan algebra and a Stone algebra (i.e., a pseudo-complemented distributive lattice satisfying the Stone identity \(\mathop {\sim }x \vee \mathop {\sim }\mathop {\sim }x \approx 1\)). IS-algebras have been studied algebraically and topologically since the 1980s, but a corresponding logic (here denoted \(\mathcal {IS}_{\le }\)) has been introduced only very recently. This logic is the departing point of the present study, which we then extend to a wide family of previously unknown logics defined from IS-algebras. We show that \(\mathcal {IS}_{\le }\) is a conservative expansion of the Belnap-Dunn four-valued logic (i.e., the order-preserving logic of the variety of De Morgan algebras), and we give a finite Hilbert-style axiomatization for it. More generally, we introduce a method for expanding conservatively every super-Belnap logic (i.e., every strengthening of the Belnap-Dunn logic) so as to obtain an extension of \(\mathcal {IS}_{\le }\). We show that every logic thus defined can be axiomatized by adding a fixed finite set of multiple-conclusion rule schemata to the corresponding super-Belnap base logic. Our results entail that the lattice of super-Belnap logics (which is known to be uncountable) embeds into the lattice of extensions of \(\mathcal {IS}_{\le }\). In fact, as in the super-Belnap case, we establish that the finitary extensions of \(\mathcal {IS}_{\le }\) are already uncountably many. When the base super-Belnap logic possesses a disjunction, we show that we can reduce the multiple-conclusion calculus to a traditional one; some of the multiple-conclusion axiomatizations so introduced are analytic and are thus of independent interest from a proof-theoretic standpoint. We also consider a few extensions of \(\mathcal {IS}_{\le }\) that cannot be obtained in the above-described way, but can nevertheless be axiomatized finitely by other methods.

Median prime ideals of pseudo-complemented distributive lattices

Coherent ideals, strongly coherent ideals, and $\tau $-closed ideals are introduced in pseudo-complemented distributive lattices and their characterization theorems are derived. A set of equivalent conditions is derived for every ideal of a pseudo-complemented distributive lattice to become a coherent ideal. The notion of median prime ideals is introduced and some equivalent conditions are derived for every maximal ideal of a pseudo-complemented distributive lattice to become a median prime ideal which leads to a characterization of Boolean algebras.

On the relation of negations in Nelson algebras

The aim of this paper is to investigate the relation between the strong and the "weak" or intuitionistic negation in Nelson algebras. To do this, we define the variety of Kleene algebras with intuitionistic negation and explore the Kalman's construction for pseudocomplemented distributive lattices. We also study the centered algebras of this variety.

Coaxer Lattices

AbstractThe notion of coaxers is introduced in a pseudo-complemented distributive lattice. Boolean algebras are characterized in terms of coaxer ideals and congruences. The concept of coaxer lattices is introduced in pseudo-complemented distributive lattices and characterized in terms of coaxer ideals and maximal ideals. Finally, the coaxer lattices are also characterized in topological terms.

Unification on Subvarieties of Pseudocomplemented Distributive Lattices

In this paper subvarieties of pseudocomplemented distributive lattices are classified by their unification type. We determine the unification type of every particular unification problem in each subvariety of pseudocomplemented distributive lattices.

NEUTRAL OPERATORS AND TRIPLE REPRESENTATION

The basic idea of triple representation of a structure is to associate two simpler structures and a connecting map. Triple representation was first introduced by Chen and Gratzer to Stone Lattices (2), (3). The concept was gen- eralised to distributive pseudo-complemented lattices and then to distributive pseudo-complemented semilattices by Katrinak. The triple representation for modular Pseudo Complemented semilattice was suceeded by W.H.Cornish(4). Now the triple representation was obtained for a supermodular semilattice and thereby the supermodular semilattices are characterised.

Determinant theory for lattice matrices

The determinant theory for matrices over a pseudo-complemented distributive lattice is presented. Previous results on this topic are special cases of the theorems proved in this paper.

Boolean Filters of Distributive Lattices

In this paper we introduce the notion of Boolean filters in a pseudo-complemented distributive lattice and characterize the class of all Boolean filters. Further a set of equivalent conditions are derived for a proper filter to become a prime Boolean filter. Also a set of equivalent conditions is derived for a pseudo-complemented distributive lattice to become a Boolean algebra. Finally, a Boolean filter is characterized in terms of congruences.

On Gentzen Relations Associated with Finite-valued Logics Preserving Degrees of Truth

When considering m-sequents, it is always possible to obtain an m-sequent calculus VL for every m-valued logic (defined from an arbitrary finite algebra L of cardinality m) following for instance the works of the Vienna Group for Multiple-valued Logics. The Gentzen relations associated with the calculi VL are always finitely equivalential but might not be algebraizable. In this paper we associate an algebraizable 2-Gentzen relation with every sequent calculus VL in a uniform way, provided the original algebra L has a reduct that is a distributive lattice or a pseudocomplemented distributive lattice. We also show that the sentential logic naturally associated with the provable sequents of this algebraizable Gentzen relation is the logic that preserves degrees of truth with respect to the original algebra (in contrast with the more common logic that merely preserves truth). Finally, for some particular logics we obtain 2-sequent calculi that axiomatize the algebraizable Gentzen relations obtained so far.

$\delta$-ideals in pseudo-complemented distributive lattices

The concept of $\delta$-ideals is introduced in a pseudo-complemented distributive lattice and some properties of these ideals are studied. Stone lattices are characterized in terms of $\delta$-ideals. A set of equivalent conditions is obtained to characterize a Boolean algebra in terms of $\delta$-ideals. Finally, some properties of $\delta$-ideals are studied with respect to homomorphisms and filter congruences.

Lee classes for pseudocomplemented semilattices, revisited

We give syntactic and semantic characterisations of the Lee classes of the varieties of pseudocomplemented semilattices and pseudocomplemented distributive lattices. The main result clarifies earlier work of K. B. Lee and of the second author.

Remarks on the Stone–Čech and Alexandroff compactifications of locales

Relations of strong inclusion are considered on pseudocomplemented distributive lattices to refine existing constructions of (Stone–Čech and Alexandroff) compactifications of frames.

Applications of Priestley duality in transferring optimal dualities

This paper illustrates how Priestley duality can be used in the transfer of an optimal natural duality from a minimal generating algebra for a quasi-variety to other generating algebras. Detailed calculations are given for the quasi-variety $$\mathbb{I}\mathbb{S}\mathbb{P}(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{4} )$$ of Kleene algebras and the quasi-varieties $$B$$ n of pseudocomplemented distributive lattices (n ≥ 1).

Adjointness Aspects of the Down-Set Functor

The down-set construction, when applied to the category of Boolean frames, can be viewed as a functor into the category of frames with frame homomorphisms subject to various conditions akin to openness. We prove that it has a right adjoint, which is then given by Booleanization, exactly for near openness and one other, closely related property; a similar result is obtained for the finitary case of pseudocomplemented distributive lattices. In addition, we present a characterization of the frames which are down-set frames of Boolean frames.

Priestley duality for demi-p-lattices

Priestley spaces of demi-p-lattices and almost-p-lattices are described in terms of Priestley spaces of pseudo-complemented distributive lattices. Characterizations of subvarieties of demi-p-lattices generated by a single finite subdirectly irreducible demi-p-lattice are given.

Semi-DeMorgan algebras

Semi-DeMorgan algebras are a common generalization of DeMorgan algebras and pseudocomplemented distributive lattices. A duality for them is developed that builds on the Priestley duality for distributive lattices. This duality is then used in several applications. The subdirectly irreducible semi-DeMorgan algebras are characterized. A theory of “partial diagrams” is developed, where properties of algebras are tied to the omission of certain partial diagrams from their duals. This theory is then used to find and give axioms for the largest variety of semi-DeMorgan algebras with the congruence extension property.

Optimal natural dualities. II: General theory

A general theory of optimal natural dualities is presented, built on the test algebra technique introduced in an earlier paper. Given that a set R R of finitary algebraic relations yields a duality on a class of algebras A = I S P ⁡ ( M _ ) \mathcal {A} = \operatorname {\mathbb {I}\mathbb {S}\mathbb {P}}( \underline {M}) , those subsets R ′ R’ of R R which yield optimal dualities are characterised. Further, the manner in which the relations in R R are constructed from those in R ′ R’ is revealed in the important special case that M _ \underline {M} generates a congruence-distributive variety and is such that each of its subalgebras is subdirectly irreducible. These results are obtained by studying a certain algebraic closure operator, called entailment, definable on any set of algebraic relations on M _ \underline {M} . Applied, by way of illustration, to the variety of Kleene algebras and to the proper subvarieties B n \mathbf {B}_{n} of pseudocomplemented distributive lattices, the theory improves upon and illuminates previous results.