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.
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Presented by J. Berman.
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Blyth, T.S., Silva, H.J. Congruence kernels in Ockham algebras. Algebra Univers. 78, 55–65 (2017). https://doi.org/10.1007/s00012-017-0441-4
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DOI: https://doi.org/10.1007/s00012-017-0441-4