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An Internal Characterization of the 0-Radical of a Semigroup

✍ Scribed by D. R. Latorre

Publisher: John Wiley and Sons
Year: 1970
Tongue: English
Weight: 156 KB
Volume: 45
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19700450115

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✦ Synopsis

In studying the algebraic structure of semigroups, H. J. HOEHNKE in [I] and [a] has used respresentations of a semigroup S by transformations on a set to introduce a radical, rad S , as a certain congruence on S , and an associated ideal rado S of S , called the 0-radical of S.

An internal characterization of the congruence r a d S was given by H. SEIDEL in [3]. The purpose of this article is to present an internal characterization of the ideal Pado S as the greatest quasi-regular ideal of S.

The relation rad" (rad" S) = rad" S then follows easily. This relation, wellknown for the JACOBSON radical of a ring, was proved for semigroups with zero by HOEHNKE and SEIDEL in [4]. However, lacking the characterization of Tado S, other methods were used there.

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