Consistent posets

Ivan Chajda; Helmut Länger

doi:10.1007/s00500-021-05906-y

Consistent posets

Soft comput. 2021;25(15):9765-9772. doi: 10.1007/s00500-021-05906-y. Epub 2021 Jun 15.

Authors

Ivan Chajda¹, Helmut Länger^{1

2}

Affiliations

¹ Department of Algebra and Geometry, Faculty of Science, Palacký University Olomouc, 17. listopadu 12, 771 46 Olomouc, Czech Republic.
² Institute of Discrete Mathematics and Geometry, Faculty of Mathematics and Geoinformation, TU Wien, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria.

Abstract

We introduce so-called consistent posets which are bounded posets with an antitone involution $^{'}$ where the lower cones of $x, x^{'}$ and of $y, y^{'}$ coincide provided that x, y are different from 0, 1 and, moreover, if x, y are different from 0, then their lower cone is different from 0, too. We show that these posets can be represented by means of commutative meet-directoids with an antitone involution satisfying certain identities and implications. In the case of a finite distributive or strongly modular consistent poset, this poset can be converted into a residuated structure and hence it can serve as an algebraic semantics of a certain non-classical logic with unsharp conjunction and implication. Finally we show that the Dedekind-MacNeille completion of a consistent poset is a consistent lattice, i.e., a bounded lattice with an antitone involution satisfying the above-mentioned properties.

Keywords: Adjointness; Antitone involution; Commutative meet-directoid; Consistent poset; Dedekind–MacNeille completion; Distributive poset; Residuation; Strongly modular poset.