Orthomodular semilattices (Q861800)

Quantum structures are usually bounded posets, but attempts were made to introduce also generalizations having only the lower bound,~\(0\). Then the orthocomplement is replaced by the relative complement, \(x^a\), of \(x\) in the interval \([0,a]\). The author studies three classes of semilattices derived from ortholattices (in descending order of generality): orthosemilattices, orthomodular semilattices, and orthosemilattices satisfying the compatibility condition \(x\leq a\leq b \Rightarrow x^a=x^b\land a\). Using a binary operation with the meaning \((x\land y)^y\) (a modification of the Sasaki projection), these three classes of semilattices can be described as equational classes.