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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.

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