Complemented Ordered Sets 1992

Abstract:  . We introduce the concept of complementary elements in ordered sets. If an ordered set S is a lattice, this concept coincides with that for lattices. The connections between distributivity and the uniqueness of complements are shown and it is also shown that modular complemented ordered sets represents "geometries" which are more general than projective planes. It was shown in [2], [4] and [6] that some of lattice properties like distributivity or modularity can be generalized also for ordered sets. We can generalize also the concept of lattice complement. The aim of this paper is to show some basic properties of complements in ordered sets and to determine connections between complementarity and distributivity. Let A be an ordered set, denote by its ordering. For M ` A, put U (M ) = fx 2 A; m x for each m 2 Mg L(M ) = fy 2 A; y m for each m 2 Mg : If M is a finite set, e.g. M = fa 1 ; : : : ; ang, we write briefly U (a 1 ; : : : ; an ) and L(a 1 ; : : : ; an ) instead of ...