Non–commutative Symmetric Differences in Orthomodular Lattices

We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the symmetric difference in Boolean algebras and four noncommutative terms. It turns out that in many respects the non-commutative forms, though more complex with respect to the lattice operations, in their properties are much nearer to the symmetric difference in Boolean algebras than the commutative terms. As application we demonstrate the usefullness of non-commutative symmetric differences in the context of congruence relations.