Symmetric Differences on Posets with an Antitone Involution

摘要

The notion of symmetric difference is defined for arbitrary posets having an antitone involution in such a way that for Boolean algebras one obtains the usual notion of symmetric difference. In the case of lattices a very natural description of symmetric differences follows. Sufficient conditions for the existence of such differences are provided. It turns out that for the existence of symmetric differences it is necessary for any two orthogonal elements to have a join. Changing the concept of symmetric difference it is also possible to define symmetric differences in directed posets with an antitone involution in a natural way such that two orthogonal elements need not have a join.