Semigroups whose any subsemigroup contains a definite element
Morio Sasaki, Reiko Inoue · Proceedings of the Japan Academy Series A Mathematical Sciences · 1964
A semigroup S is called a -semigroup if S satisfies the follow- ing two conditions"(1) Any subset of S which contains a definite element e is a subsemigroup of S.(2) Any subsemigroup of S contains e.Recently T. Tamura 5 has determined all the types of -semi- groups and one of the authors 3 has done the construction of semigroups which satisfy (1).In this paper, we shall investigate the semigroups satisfying (2).Such semigroups are called *-semigroups.A finite unipotent semigroup is a *-semigroup.Let S be a *-semigroup and e be a definite element of S. Lemma 1.A subsemigroup of a *-semigroup is a *-semigroup.Lemma 2. A homomorphic image of a *-semigroup is a *semigroup.