Congruence Kernels of Orthoimplication Algebras

Abstracting from certain properties of the implication operation in Boolean algebras leads to so-called orthoimplication algebras. These are in a natural one-to-one correspondence with families of compatible orthomodular lattices. It is proved that congruence kernels of orthoimplication algebras are in a natural one-to-one correspondence with families of compatible p-filters on the corresponding orthomodular lattices. Finally, it is proved that the lattice of all congruence kernels of an orthoimplication algebra is relatively pseudocomplemented and a simple description of the relative pseudocomplement is given. In the literature many attempts were made in order to investigate properties of the implication operation in generalizations of Boolean algebras. These attempts led to different types of so-called implication algebras (cf. e. g. [2], [5] and [6]). It is interesting to note that these types of implication algebras are in a natural one-to-one correspondence with join-semilattices with 1 the principal filters of which are certain generalizations of Boolean algebras. Hence the question arises if there is a natural one-to-one correspondence between congruence kernels of these implication algebras on the one side and certain families of congruence kernels of the corresponding generalizations of Boolean algebras on the other side. We solve this problem for so-called orthoimplication algebras introduced in [2]. Moreover, we prove that the lattice of congruence kernels of orthoimplication algebras is relatively pseudocomplemented and we derive a simple description of the relative pseudocomplement. In [1] implication algebras were introduced as algebras reflecting properties of the implication operation in Boolean algebras: Definition 1. (cf. [2]) An orthoimplication algebra is an algebra (A, ·, 1) of type (2, 0) satisfying xx = 1, (xy)x = x, (xy)y = (yx)x, x((yx)z) = xz. Received March 16, 2006. 2000 Mathematics Subject Classification. Primary 08A30, 20N02, 06A12, 06C15.