Algebras describing pseudocomplemented, relatively pseudocomplemented and sectionally pseudocomplemented posets

TL;DR

This paper develops algebraic frameworks for various classes of pseudocomplemented posets, enabling their analysis through universal algebra methods and revealing strong congruence properties with applications in non-classical logics.

Contribution

It introduces algebraic characterizations for pseudocomplemented posets using commutative directoids and lambda-lattices, establishing their properties and applications.

Findings

Algebras fully characterize pseudocomplemented posets.

Assigned algebras satisfy strong congruence properties.

Applications in non-classical logics are discussed.

Abstract

In order to be able to use methods of Universal Algebra for investigating posets, we assign to every pseudocomplemented poset, to every relatively pseudocomplemented poset and to every sectionally pseudocomplemented poset a certain algebra (based on a commutative directoid or on a lambda-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. It turns out that the assigned algebras satisfy strong congruence properties which can be transferred back to the posets. We also mention applications of such posets in certain non-classical logics.