ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. 65,   2   (1996)
pp.   247-279

GENERALIZED DIFFERENCE POSETS AND ORTHOALGEBRAS
J. HEDLIKOVA and S. PULMANNOVA

Abstract.  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 conditions $a\leq b \implies b\ominus (b\ominus a) = a$ and $a\leq b\leq c \implies (c\ominus a) \ominus(c\ominus b) = b\ominus a$. 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 $b\ominus a = c\ominus a \implies b = c$. 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.

AMS subject classification.  06A06, 08A55, 06C15
Keywords.  Difference (operation) on a poset, orthomodular poset, orthoalgebra, difference poset, order ideal, orthogonality relation, sum (operation) on a set, orthomodular group