Consistent posets
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 x,y are different form 0,1 and, moreover, if x,y are different form 0 then their lower cone is different form 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.
Cite
@article{arxiv.2006.16053,
title = {Consistent posets},
author = {Ivan Chajda and Helmut Länger},
journal= {arXiv preprint arXiv:2006.16053},
year = {2020}
}
Comments
arXiv admin note: text overlap with arXiv:2006.04417