Generalized difference posets and orthoalgebras.

A diierence on a poset (P;) is a partial binary operation on P such that b a is deened if and only if a b subject to conditions a b =) b (b a) = a and a b c =) (c a) (c b) = b a. A diierence poset (DP) is a bounded poset with a diierence. A generalized diierence poset (GDP) is a poset with a diierence having a smallest element and the property b a = c a =) b = c. We prove that every GDP is an order ideal of a suitable DP, thus extendingprevious similar results of Janowitz for generalizedorthomodular lattices and of Mayet-Ippolito for (weak) generalized orthomodular posets. Various results and examples concerning posets with a diierence are included. 0. Introduction A diierence (operation) on a partially ordered set (poset) P is a partial binary operation on P such that b a is deened if and only if a b satisfying some conditions. For example, b ^ a 0 is such an operation in an orthomodular poset. A diierence poset (DP) is a bounded poset equipped with a diierence operation. For example, every orthoalgebra (which is a natural generalization of an orthomodular lattice or poset) is a diierence poset. An introduction to diierence posets is in K, Ch]. A basic theory of orthoal-gebras can be found in F, G, R]. An orthoalgebra (OA) is deened as a partial binary algebra with a sum (operation) on a set with two special elements. An exact relationship between diierence posets and orthoalgebras was pointed out in N, P]. A description of an orthoalgebra in terms of a diierence operation on a poset is given there. A description of an orthomodular poset, resp. a diier-ence poset in terms of a sum operation on a set is given in B, M], resp. P] and F, B]. Yet more general approach is used in K, R] when considering a diierence operation on an arbitrary set with a special element. We deene a generalized diierence poset (GDP) as a poset with a smallest element and with a diierence operation subject to an additional condition in