Consistent posets

doi:10.1007/s00500-021-05906-y

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Consistent posets

2021-06-15

Ivan Chajda

Helmut Länger *

DOI：10.1007/s00500-021-05906-y

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Abstract

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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:

Consistent poset

Antitone involution

Distributive poset

Strongly modular poset

Commutative meet-directoid

Residuation

Adjointness

Dedekind-MacNeille completion

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