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.
Highlights
In some non-classical logics the contraposition law is assumed
An algebraic semantics of such logics is provided by means of De Morgan posets, i.e., bounded posets equipped with a unary operation which is an antitone involution
Support of the research of the authors by the Austrian Science Fund (FWF), project I 4579-N, and the Czech Science Foundation (GAC R), Project 20-09869L, entitled “The many facets of orthomodularity,” as well as by ÖAD, project CZ 02/2019, entitled “Function algebras and ordered structures related to logic and data fusion,” and, concerning the first author, by IGA, project PrF 2021 030, is gratefully acknowledged
Summary
In some non-classical logics the contraposition law is assumed. An algebraic semantics of such logics is provided by means of De Morgan posets, i.e., bounded posets equipped with a unary operation which is an antitone involution. We can release the assumption that x, y are comparable, but, on the other hand, we will ask that L(x, y) = {0} if and only if at least one of the entries x, y is equal to 0 Such a poset will be called consistent in the sequel. We will define operators assigning to the couple x, y of entries a certain subset of P It is in accordance with the description of uncertainty of such a logic based on the fact that a poset instead of a lattice is used. The poset P is called bounded if it has a least element 0 and a greatest element 1 This fact will be expressed by notation (P, ≤, 0, 1).
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