2020/06/26 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1
Computer Science · #03G25 #06A11 #06B75 #06C15 #Advanced Algebra and Logic #FOS: Mathematics #Logic (math.LO) #Logic, Reasoning, and Knowledge #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2006.16053
openalex publication_date 2020/06/26 · openalex created_date 2022/09/30 · openalex updated_date 2026/07/28
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.