Algebraic Aspects of Relatively Pseudocomplemented Posets

Ivan Chajda,Helmut Länger,Jan Paseka

doi:10.1007/s11083-019-09488-1

Ivan Chajda, Helmut Länger + Show 1 more

Open Access

https://doi.org/10.1007/s11083-019-09488-1

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Journal: Order	Publication Date: May 1, 2019
Citations: 7	License type: open-access

Affiliation: Palacký University, Olomouc, TU Wien, Masaryk University

Abstract

In Chajda and Länger (Math. Bohem. 143, 89–97, 2018) the concept of relative pseudocomplementation was extended to posets. We introduce the concept of a congruence in a relatively pseudocomplemented poset within the framework of Hilbert algebras and we study under which conditions the quotient structure is a relatively pseudocomplemented poset again. This problem is solved e.g. for finite or linearly ordered posets. We characterize relative pseudocomplementation by means of so-called L-identities. We investigate the category of bounded relatively pseudocomplemented posets. Finally, we derive certain quadruples which characterize bounded Hilbert algebras and bounded relatively pseudocomplemented posets up to isomorphism using Glivenko equivalence and implicative semilattice envelope of Hilbert algebras.

Highlights

The origin of pseudocomplemented and relatively pseudocomplemented lattices and semilattices can be found in papers quoted in the references in the monographs by H.B
In what follows we show that θ a is a principal congruence and that every principal congruence on a Hilbert algebra is of this form
In the sequel we show that the Dedekind-MacNeille completion of a relatively pseudocomplemented poset is again relatively pseudocomplemented

Summary

Introduction

The origin of pseudocomplemented and relatively pseudocomplemented lattices and semilattices can be found in papers quoted in the references in the monographs by H.B. We cannot omit the paper [33] by S Rudeanu where he proves that every relatively pseudocomplemented poset is a Hilbert algebra w.r.t. relative pseudocomplementation. A homomorphism of (bounded) relatively pseudocomplemented posets is a homomorphism of (bounded) Hilbert algebras. Definition 1.2 An implicative semilattice is an algebra (A, ∧, ∗, 1) of type (2, 2, 0) such that (A, ∧) is a meet-semilattice and (A, ∗, 1) is a relatively pseudocomplemented poset. The following theorem summarizes some properties from [5], of interest for the present work: Theorem 1.3 There exists a functor S : H → IS that maps every Hilbert algebra A to an implicative semilattice S(A), and every homomorphism h : A → B of Hilbert algebras to a homomorphism S(h) : S(A) → S(B) of implicative semilattices such that S(h)( X) =.

Congruences

The Dedekind-MacNeille Completion and Generalized Glivenko Equivalences

The Category of Bounded Relatively Pseudocomplemented Posets

Characterizing Quadruples