Logical and algebraic properties of generalized orthomodular posets (Q2122477)

The authors investigate logical and algebraic properties of generalized orthomodular posets, i.e., orthoposets \((P, \le, {'}, 0, 1)\) fulfilling the condition \(x \le y\) implies \(y = x \lor L(x',y)\), where \(L(x',y)\) is the set of lower bounds of \(\{x', y\}\) (and \(x \lor L(x',y) = \{ x \lor z : z \in L(x',y)\}\)). Some other notions are defined analogously using the properties of sets of upper and/or lower bounds generalizing algebraic identities. In particular, they study the possibility to convert such posets into operator residuated structures, representation by means of algebras with everywhere defined operations (and congruence properties for the class of such algebras), Dedekind-MacNeille completion.