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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A network of congruences on an inverse semigroup

Authors: Mario Petrich and Norman R. Reilly
Journal: Trans. Amer. Math. Soc. 270 (1982), 309-325
MSC: Primary 20M10
MathSciNet review: 642343
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Abstract: A congruence $\rho$ on an inverse semigroup $S$ is determined uniquely by its kernel and its trace. Denoting by ${\rho ^{\min }}$ and ${\rho _{\min }}$ the least congruence on $S$ having the same kernel and the same trace as $\rho$, respectively, and denoting by $\omega$ the universal congruence on $S$, we consider the sequence $\omega$, ${\omega ^{\min }}$, ${\omega _{\min }}$, ${({\omega ^{\min }})_{\min }}$, ${({\omega _{\min }})^{\min }} \ldots$. These congruences, together with the intersections of corresponding pairs, form a sublattice of the lattice of all congruences on $S$. We study the properties of these congruences and establish several properties of the quasivarieties of inverse semigroups induced by them.

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Keywords: Inverse semigroups, congruences, lattice of congruences, implications, kernel and trace of a congruence
Article copyright: © Copyright 1982 American Mathematical Society