Boolean D-posets. (English) Zbl 0915.03052

A D-poset is an algebraic structure \(\mathcal P\) which is a poset with least and greatest elements \(0\) and \(1\), respectively, and with a partial operation \(\setminus \) such that \( a\setminus b\) is defined iff \(b \leq a\), and the following axioms for \(\setminus \) hold: (i) if \(a \leq b,\) then \(b \setminus a \leq a\) and \(b \setminus (b \setminus a) = a\); (ii) if \(a\leq b \leq c\), then \(c \setminus b \leq c \setminus a\) and \((c \setminus a) \setminus ( c \setminus b) = b \setminus a.\) If \(\mathcal P\) is a lattice such that \(a \setminus (a \wedge b) = (a \vee b) \setminus b\) for any \(a,b \in \mathcal P\), \(\mathcal P\) is said to be a Boolean D-poset. We recall that Boolean D-posets are categorically equivalent to MV-algebras.
In addition, the notion of the compatibility and ideals of D-posets are studied.


03G12 Quantum logic