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Congruences on regular semigroups


Authors: Francis Pastijn and Mario Petrich
Journal: Trans. Amer. Math. Soc. 295 (1986), 607-633
MSC: Primary 20M10; Secondary 08A30, 20M17
DOI: https://doi.org/10.1090/S0002-9947-1986-0833699-3
MathSciNet review: 833699
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Abstract: Let $ S$ be a regular semigroup and let $ \rho $ be a congruence relation on $ S$. The kernel of $ \rho $, in notation $ \ker \rho $, is the union of the idempotent $ \rho $-classes. The trace of $ \rho $, in notation $ \operatorname{tr}\,\rho $, is the restriction of $ \rho $ to the set of idempotents of $ S$. The pair $ (\ker \rho ,\operatorname{tr}\,\rho )$ is said to be the congruence pair associated with $ \rho $. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple $ ((\rho \vee \mathcal{L})/\mathcal{L},\ker \rho ,(\rho \vee \mathcal{R})/\mathcal{R})$ is said to be the congruence triple associated with $ \rho $. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple.

The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of $ S$. For congruence relations $ \rho $ and $ \theta $, put $ \rho {T_l}\theta \;[\rho {T_r}\theta ,\rho T\theta ]$ if and only if $ \rho \vee \mathcal{L} = \theta \vee \mathcal{L}\;[\rho \vee \mathcal{R} = \theta \vee \mathcal{R},\operatorname{tr}\rho = \operatorname{tr}\theta ]$. Then $ {T_l},{T_r}$ and $ T$ are complete congruences on the congruence lattice of $ S$ and $ T = {T_l} \cap {T_r}$.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0833699-3
Article copyright: © Copyright 1986 American Mathematical Society

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