# 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

, Volume 78 (1) – May 22, 2017
11 pages

/lp/springer_journal/congruence-kernels-in-ockham-algebras-s5dIeapStV
Publisher
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

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations

Abstract access only

18 million full-text articles

Print

20 pages / month

PDF Discount

20% off