Congruence kernels in Ockham algebras

Abstract

We consider, in the context of an Ockham algebra ${{\mathcal{L} = (L; f)}}$, the ideals I of L that are kernels of congruences on ${\mathcal{L}}$. We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel ${I \neq L}$ is the intersection of the prime ideals P such that ${I \subseteq P}$, ${P \cap f(I) = \emptyset}$, and ${f^{2}(I) \subseteq P}$. The congruence kernels form a complete lattice which in general is not modular. For every non-empty subset X of L, we also describe the smallest congruence kernel to contain X, in terms of which we obtain necessary and sufficient conditions for modularity and distributivity. The case where L is of finite length is highlighted.