Algebras Describing Pseudocomplemented, Relatively Pseudocomplemented and Sectionally Pseudocomplemented Posets

Ivan Chajda,Helmut Länger

doi:10.3390/sym13050753

Abstract

In order to be able to use methods of universal algebra for investigating posets, we assigned 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 λ-lattice) which satisfies certain identities and implications. We show that the assigned algebras fully characterize the given corresponding posets. A certain kind of symmetry can be seen in the relationship between the classes of mentioned posets and the classes of directoids and λ-lattices representing these relational structures. As we show in the paper, this relationship is fully symmetric. Our results show 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.

Highlights

When investigating algebras, researchers usually apply well-known algebraic methods and results
We believe that our approach can bring new insight into the study of pseudocomplemented, relatively pseudocomplemented and sectionally pseudocomplemented posets, since we provide a purely algebraic description of them, enabling the application of algebraic tools for their investigation
A relatively pseudocomplemented poset P is determined by its assigned algebra A which is a commutative meet-directoid with constant 1 and equipped with a binary operation ∗

Summary

Introduction

Researchers usually apply well-known algebraic methods and results This does not work in the case of partially ordered sets (posets). Quackenbush [1] showed that if a poset P is up-directed or down-directed, a certain algebra with one binary operation, a so-called directoid, can be assigned to P. This assignment is in general not unique, but, from every such assigned directoid, P can be reconstructed in a unique way. The class of directoids assigned to such posets forms a variety of algebras This is of great advantage since there exist many methods and results for studying varieties in general algebra. As we show in the paper, this relationship is fully symmetric

Preliminaries

Pseudocomplemented Posets

Relatively Pseudocomplemented Posets

Sectionally Pseudocomplemented Posets

Congruence Properties

Conclusions and Discussion