Vol. 33, No. 1, 1970

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On a regular semigroup in which the idempotents form a band

Miyuki Yamada

Vol. 33 (1970), No. 1, 261–272

This paper is a continuation of a previous paper, in which the structure of certain regular semigroups, called generalized inverse semigroups, has been studied. A semigroup is called strictly regular if it is regular and the set of all its idempotents is a subsemigroup. A generalized inverse semigroup is strictly regular, but the converse is not true. Hence, the class of generalized inverse semigroups is properly contained in the class of strictly regular semigroups. The main purpose of this paper is to establish some results which clarify the structure of strictly regular semigroups. The concept of a quasi-direct product of a band (that is, an idempotent semigroup) and an inverse semigroup is introduced, and in particular it is proved that any semigroup is strictly regular if and only if it is a quasi-direct product of a band and an inverse semigroup.

Mathematical Subject Classification
Primary: 20.93
Received: 12 May 1969
Published: 1 April 1970
Miyuki Yamada