Dorfer, G. (1998). Kongruenzen und symmetrische Differenzen auf orthomodularen Verbänden [Dissertation, Technische Universität Wien]. reposiTUm. https://resolver.obvsg.at/urn:nbn:at:at-ubtuw:1-11308
In this dissertation congruence relations and symmetric differences in orthomodular lattices are studied. After preliminaries in chapter 1 the first two sections of chapter 2 deal with so-called p-ideals, which are exactly the congruence kernels. Amongst other things infimum, supremum, pseudocomplement and relative pseudocomplement in the congruence lattice is explicitly described with p-ideals. In section 2.3 congruence classes are characterized and represented by means of the corresponding p-ideal. From these results congruence regularity, -permutability and -uniformity for orthomodular lattices is derived in 2.4. In chapter 3 we first determine all possibilities to define a symmetric difference in an orthomodular lattice by a term function and then investigate these operations. It turns out that 4 operations out of the 6 that emerged are one-sided regular and one-sided invertible while the other 2 meet regularity or invertibility conditions for Boolean algebras only. Furthermore associativity and distributivity of symmetric differences with respect to meet operations is studied. The main result here is that one of these identities holds if and only if the orthomodular lattice is a Boolean algebra.