New Operations on Orthomodular Lattices: ``Disjunction'' and ``Conjunction'' Induced by Mackey Decompositions

Abstract

New conjunctionlike and disjunctionlike operations on orthomodular lattices are defined with the aid of formal Mackey decompositions of not necessarily compatible elements. Various properties of these operations are studied. It is shown that the new operations coincide with the lattice operations of join and meet on compatible elements of a lattice but they necessarily differ from the latter on all elements that are not compatible. Nevertheless, they define on an underlying set the partial order relation that coincides with the original one. The new operations are in general nonassociative: if they are associative, a lattice is necessarily Boolean. However, they satisfy the Foulis-Holland-type theorem concerning associativity instead of distributivity.