An algebraic axiomatization of orthogonal posets.
An orthogonal poset is a bounded poset \(\mathcal P=(P,\leq,{'},0,1)\) with an antitone involution such that \(x\vee y\) exists whenever \(x\leq y'\) and satisfying the identity \(x\vee x'\approx 1\). An orthocomplemented directoid is an algebra \(\mathcal D=(D,+,{'},0,1)\) of type \((2,1,0,0)\) satisfying the identities \(x+x\approx x\), \(x+y\approx y+x\), \(x+((x+y)+z)\approx(x+y)+z\), \((x')'\approx x\), \(x+0\approx x\), \(x+x'\approx 1\) and \((((x+z)+(y+z)')'+(y+z)')+z'\approx z'\). It is proved that if \(\mathcal D\) is an orthocomplemented directoid and one defines \(x\leq y\) if \(x+y=y\) (\(x,y\in D\)) then \(\mathbb P(\mathcal D):=(D,\leq,{'},0,1)\) is an orthogonal poset. Conversely, if \(\mathcal P\) is an orthogonal poset and one defines \(x+y:=x\vee y\) if \(x\vee y\) exists and \(x+y\) to be an arbitrary upper bound of \(x\) and \(y\) otherwise (\(x,y\in P\)) then \(\mathbb D(\mathcal P):=(P,+,{'},0,1)\) is an orthocomplemented directoid. We have \(\mathbb P(\mathbb D(\mathcal P))=\mathcal P\). Moreover, the variety of orthocomplemented directoids is congruence distributive.
scientific article; zbMATH DE number 6506243