, Volume 3, Issue 1, pp 160-167

Note on exclusive semigroups

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In the SEMIGROUP FORUM, Vol. 1, No. 1, B. M. Schein proposed the following problem:

Describe the structure of semigroups S such that for every a,b,c∈S, abc=ab, bc or ac. At present, we shall call such a semigroup S anexclusive semigroup. Recently, the author heard that the structure of commutative exclusive semigroups was completely determined by T. Tamura [3]. In this paper, we deal with exclusive semigroups which are not necessarily commutative. The paper is divided into three sections. At first, the structure of exclusive semigroups whose idempotents form a rectangular band will be clarified. Next, we shall investigate a certain class of exclusive semigroups called “exclusive homobands”. Especially, in the final section we shall deal with medial exclusive homobands and show how to construct them. The proofs are omitted and will be given in detail elsewhere.

Communicated by T. Saito