We introduce so-called consistent posets which are bounded posets with an antitone involution ′\documentclass[12pt]{minimal}
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\begin{document}$$'$$\end{document} where the lower cones of x,x′\documentclass[12pt]{minimal}
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\begin{document}$$x,x'$$\end{document} and of y,y′\documentclass[12pt]{minimal}
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\begin{document}$$y,y'$$\end{document} 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.
