Some Elementary Properties of Conditionally Distributive Lattices (original) (raw)

Distributive laws for concept lattices

Algebra Universalis, 1993

We study several kinds of distributivity for concept lattices of contexts. In particular, we find necessary and sufficient conditions for a concept lattice to be (1) distributive, (2) a frame (locale, complete Heyting algebra), (3) isomorphic to a topology, (4) completely distributive, (5) superalgebraic (i.e., algebraic and completely distributive). In cases (2), (4) and (5), our criteria are first order statements on objects and attributes of the given context. Several applications are obtained by considering the completion by cuts and the completion by lower ends of a quasiordered set as special types of concept lattices. Various degrees of distributivity for concept lattices are expressed by certain "separation axioms" for the underlying contexts. Passing to complementary contexts makes some statements and proofs more elegant. For example, it leads to a one-to-one correspondence between completely distributive lattices and so-called Cantor lattices, and it establishes an equivalence between partially ordered sets and doubly founded reduced contexts with distributive concept lattices.

Bounded distributive lattices with strict implication

Mathematical Logic Quarterly, 2005

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Lattices and Their Consistent Quantification

Annalen der Physik, 2018

This paper introduces the order-theoretic concept of lattices along with the concept of consistent quantification where lattice elements are mapped to real numbers in such a way that preserves some aspect of the ordertheoretic structure. Symmetries, such as associativity, constrain consistent quantification, and lead to a constraint equation known as the sum rule. Distributivity in distributive lattices also constrains consistent quantification and leads to a product rule. The sum and product rules, which are familiar from, but not unique to, probability theory, arise from the fact that logical statements form a distributive (Boolean) lattice, which exhibits the requisite symmetries.

Distributive Lattices with a Generalized Implication: Topological Duality

A Journal on The Theory of Ordered Sets and Its Applications, 2011

In this paper we introduce the notion of generalized implication for lattices, as a binary function ⇒ that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ⇒ defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.

Two problems about perfect distributive lattices

Archiv der Mathematik, 1987

The study of the class P of perfect distributive lattices has been initiated in [2]. In this note we solve two problems left open in [2]. In the first section we give two subdirect representation theorems for the class Pol of perfect bounded distributive lattices (Theorems 1.4 and 1.8) and in the second one we determine those posets which are representable over P (Theorem 2.5). 0. Preliminaries. We assume familiarity with the elements of sheaf theory and the Priestley duality theory. For these topics we refer to [1] and [8], [5] and [6] respectively. Nevertheless, in order to make this paper more or less self-contained, we recall the main definitions and properties needed. All lattices that we consider in this paper are distributive and therefore the adjective "distributive" will generally be omitted. The notation D stands for the class of alldistributive-lattices. The subscript 0 (resp. 1) means that the lattices under consideration are bounded below (resp. above). A lattice Lisperfect if its prime spectrum, Spec L, (i. e., the set of its prime ideals ordered by inclusion) is the cardinal sum of chains. The class of all perfect lattices is denoted by P and the symbols P0, P1, P01 have an obvious meaning. The basic facts about perfect lattices are contained in [2]. A Boolean product (resp. weak Boolean product) of a family (A x [ x ~ X) of algebras over a Boolean space X is a subdirect product A of the given family such that the following conditions hold: (i) if a, b ~ A, then ~a = b~ = {x]ax = b~} is clopen (resp. open) in X; (it) if a, beA and Wis clopen in X, then a]w w b]_w~A. Obviously the definition of a weak Boolean product (resp. Boolean product) corresponds to sheaves (resp. Hausdorff sheaves) of algebras over a Boolean space [8]. A weak Boolean representation of a lattice L is an isomorphism O from L onto a weak Boolean product of lattices. As noticed in [3], we may always assume that the weak Boolean representations of non-trivial lattices are proper, that is, no stalk is trivial.

On the content of lattices of logics. Part I

For every consequence (or closure) operator Cn on a set S, the family C of all Cn-closed sets, partially ordered by set inclusion, forms a complete lattice, called the lattice of logics. If the lattice C is distributive, then, it forms a Heyting algebra, since it has the zero element Cn(0) and is complete. Logics determined by this Heyting algebra is studied in the second part. In Part I it is shown (for Cn finitary or compact) that the lattice C is distributive iff its dual space is topological. Moreover a representation theorem for lattices of logics is given.

Some nonstandard methods applied to distributive lattices

Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1990

In memory of Abraham Robinson, on the occasion of his 70th birthday Distributive lattices are locally finite algebras, i.e. their finitely generated sublattices are finite. The general theory of finite distributive lattices is remarkably transparent, due to one or both of the following principles: (i) Any filter (or ideal) is principal. (ii) The lattice is join-generated by its join-irreducible elements, namely by those elements z for which x v y = z implies x = z or y = z .

Characterization theorem of lattice implication

In this paper we show the characterization theorem of lattice implication algebras which is presented by in 1993. Our theorem means that the class of all lattice implication algebras coincides with the class of all bounded commutative BCK-algebras and hence it is categorically equivalent to the class of M V -algebras and to the class of