Generalized difference posets and orthoalgebras. (English) Zbl 0922.06002

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.


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