Congruence kernels in Ockham algebras

Congruence kernels in Ockham algebras We consider, in the context of an Ockham algebra $${{\mathcal{L} = (L; f)}}$$ L = ( L ; f ) , the ideals I of L that are kernels of congruences on $${\mathcal{L}}$$ L . We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel $${I \neq L}$$ I ≠ L is the intersection of the prime ideals P such that $${I \subseteq P}$$ I ⊆ P , $${P \cap f(I) = \emptyset}$$ P ∩ f ( I ) = ∅ , and $${f^{2}(I) \subseteq P}$$ f 2 ( I ) ⊆ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png algebra universalis Springer Journals

Congruence kernels in Ockham algebras

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Publisher
Springer International Publishing
Copyright
Copyright © 2017 by Springer International Publishing
Subject
Mathematics; Algebra
ISSN
0002-5240
eISSN
1420-8911
D.O.I.
10.1007/s00012-017-0441-4
Publisher site
See Article on Publisher Site

Abstract

We consider, in the context of an Ockham algebra $${{\mathcal{L} = (L; f)}}$$ L = ( L ; f ) , the ideals I of L that are kernels of congruences on $${\mathcal{L}}$$ L . We describe the smallest and the largest congruences having a given kernel ideal, and show that every congruence kernel $${I \neq L}$$ I ≠ L is the intersection of the prime ideals P such that $${I \subseteq P}$$ I ⊆ P , $${P \cap f(I) = \emptyset}$$ P ∩ f ( I ) = ∅ , and $${f^{2}(I) \subseteq P}$$ f 2 ( I ) ⊆ 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.

Journal

algebra universalisSpringer Journals

Published: May 22, 2017

References

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