Generalized difference posets and orthoalgebras.
Jarmila Hedlı́ková, Sylvia Pulmannová · 1996
. A difference on a poset (P; ) is a partial binary operation \\Psi on P such that b \\Psi a is defined if and only if a b subject to conditions a b =) b \\Psi (b \\Psi a) = a and a b c =) (c \\Psi a) \\Psi (c \\Psi b) = b \\Psi 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 \\Psi a = c \\Psi a =) 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. 0. Introduction A difference (operation) on a partially ordered set (poset) P is a partial binary operation \\Psi on P such that b \\Psi a is defined if and only if a b satisfying some conditions. For example, b a 0 is such an operation in an orthomodular poset. A...