One-sided congruences on inverse semigroups

Meakin, John

doi:10.1090/S0002-9947-1975-0369580-9

One-sided congruences on inverse semigroups
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by John Meakin PDF

Trans. Amer. Math. Soc. 206 (1975), 67-82 Request permission

Abstract:

By the kernel of a one-sided (left or right) congruence $\rho$ on an inverse semigroup $S$, we mean the set of $\rho$-classes which contain idempotents of $S$. We provide a set of independent axioms characterizing the kernel of a one-sided congruence on an inverse semigroup and show how to reconstruct the one-sided congruence from its kernel. Next we show how to characterize those partitions of the idempotents of an inverse semigroup $S$ which are induced by a one-sided congruence on $S$ and provide a characterization of the maximum and minimum one-sided congruences on $S$ inducing a given such partition. The final two sections are devoted to a study of indempotent-separating one-sided congruences and a characterization of all inverse semigroups with only trivial full inverse subsemigroups. A Green-Lagrange-type theorem for finite inverse semigroups is discussed in the fourth section.

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