Consistent posets

We introduce so-called consistent posets which are bounded posets with an antitone involution $$'$$ ′ where the lower cones of $$x,x'$$ x , x ′ and of $$y,y'$$ 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.

Effect algebras, witness pairs and observables

The category of effect algebras is the Eilenberg-Moore category for themonad arising from the free-forgetful adjunction between categories of bounded posetsand orthomodular posets.In the category of effect algebras, an observable is a morphism whose domainis a Boolean algebra. The characterization of subsets of ranges of observables isan open problem.For an interval effect algebra E, a witness pair for a subset of S is an objectliving within E that "witnesses existence" of an observable whose rangeincludes S. We prove that there is an adjunction between the poset of allwitness pairs of E and the category of all partially inverted E-valuedobservables.