Generalized difference posets and orthoalgebras. (English) Zbl 0922.06002

Hedlíková, J.; Pulmannová, S.

Acta Math. Univ. Comen., New Ser. 65, No. 2, 247-279 (1996).

Summary: A difference on a poset \((P, \leq)\) is a partial binary operation \(\ominus\) on \(P\) such that \(b\ominus a\) is defined if and only if \(a \leq b\) subject to the conditions \(a \leq b \Rightarrow b\ominus (b\ominus a) = a\) and \(a \leq b \leq c \Rightarrow (c\ominus a)\ominus (c\ominus b) = b \ominus a\). A difference poset (DP) is a bounded poset with a difference. A generalized difference poset (GDP) is a poset with a difference having a smallest element and the property \(b\ominus a = c\ominus a \Rightarrow b = c\). We prove that every GDP is an order ideal of a suitable DP, thus extending previous similar results of Janowitz for generalized orthomodular lattices and of Mayet-Ippolito for (weak) generalized orthomodular posets. Various results and examples concerning posets with a difference are included.

MSC:

06A06	Partial orders, general
03G12	Quantum logic
06C15	Complemented lattices, orthocomplemented lattices and posets
08A55	Partial algebras

Keywords:

orthomodular posets; order ideal; orthogonality relation; orthomodular group; difference poset; generalized difference poset