On Inversible Semigroups

Abstract

Let S be a semigroup, and let I be the totality of all idempotents of S.
Then S is said to be inversible if S satisfies the following two conditions ; ( 1 ) to each a∈S there exists a^[*] ∈S such that aa^[*] = a^[*]a∈ I ; (2) I is a subsemigroup of S. For instante, idempotent semigroups ( accordingly completely non-commutative semigorups) [3] [4], left ( right ) regular and right (left ) simple semigroups [2] and commutative inverse semigrourps [5] are clearly inversible semigroups.
T. Tamura showed that if I is corsisting of only one idempotent ( he defined such a semigroup to be an 'unipotent semigroup') S has the minimcal two sided ideal K ( Suschkewitsch kernel [7] )which is the same as the maximal subgroup of S. Moreover, under the same restriction he points out that the Rees factor semigroup Z= S/K [6] is a zero-semigroup and that the structure of S is completely determined by K,Z and a ramified homomorphism f of Z into K [8] .
The main purpose of this paper is to show, among other things, that the above-mentioned Tamura's results are extended to an inversible semigroup whose idempotents are primitive.
Throughout the whole paper the operation +^^[・] ([・]∑) will denote the class sum, i.e. , disjoint sum of sets.