An equational axiomatization of Post Almost Distributive Lattices (original) (raw)

Generalized Almost Distributive Lattices-I

Southeast Asian Bulletin of Mathematics, 2009

The concept of a GADL as a generalization of an ADL is introduced. Necessary and sufficient conditions for a GADL to become a distributive lattice and a GADL to become an ADL are obtained. We also study the maximal sets in a GADL and give equivalent conditions for a GADL to become a distributive lattice in terms of maximal sets.

• Dual Psuedo-Complemented Almost Distributive Lattices

International Journal of Mathematical Archive, 2012

The concept of a dual pseudo-complemented Almost Distributive Lattice is introduced. Necessary and sufficient conditions for an Almost Distributive Lattice to become a dual pseudo-complemented Almost Distributive Lattice are derived. It is proved that a dual pseudo-complemented Almost Distributive Lattice is equationally definable. A one to one correspondence between the set of all dual pseudo-complementations on an ADL and the set of all maximal elements of is obtained. Also proved that the set is a Boolean algebra.

Some Elementary Properties of Conditionally Distributive Lattices

1983

The notion of a conditionally distributive lattice was introduced by B. Wolniewicz while formally investigating the ontology of situations (cf. [2]). In several of this lectures he has appealed for a study of that class of lattices. The present abstract is a response to that request.

Chapter VI: Finite Distributive Lattices

1983

We continue our study of the finite ideals of 2 in this chapter by showing that every finite distributive lattice is isomorphic to an ideal of Q>. This result is proved using techniques extending those introduced in Chap. V. Different trees are used, and we introduce tables which provide reduction procedures from the top degree of the ideal; these tables are obtained from representations of distributive lattices. As an application, we show that the set of minimal degrees forms an automorphism base for 2. Many of the applications which we obtain in later chapters from the complete characterization of the countable ideals of 2 can be obtained from the fact that all countable distributive lattices are isomorphic to ideals of 2. We use Exercise 4.17 of this chapter to indicate how to obtain the characterization of distributive ideals of 2. This exercise allows the reader to proceed directly to Chap. VIII.2 from the end of this chapter. The results of Appendix B.I are needed for this chapter.

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 .

Equational classes of distributive double p-algebras

Algebra Universalis, Vol. 14, 235-243, 1982

An algebra (A, A, V, --1, --, 0, 1) is a double p-algebra if (A, A, V, 0, 1) is a bounded lattice, -1 is a pseudocomplementation operator and -a dual pseudocomplementation operator.

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)