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Note on exclusive semigroups

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Abstract

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.

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References

  1. CHRISLOCK, J. L.: Medial semigroups, Journal of Algebra12 (1969), 1–9.

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  2. CLIFFORD, A. H. and G. B. PRESTON: The algebraic theory of semigroups, Vol. 1, Survey 7, Amer. Math. Soc., Providence, R.I., 1961.

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  3. TAMURA, T.: On commutative exclusive semigroups, Vol. 1, Survey 7,.

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  4. YAMADA, M. and N. KIMURA: Note on idempotent semigroups, II, Proc. Japan Acad.34 (1958), 110–112.

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Authors and Affiliations

  1. Shimane University, Matsue, Shimane, Japan

    Miyuki Yamada

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  1. Miyuki Yamada
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Communicated by T. Saito

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Yamada, M. Note on exclusive semigroups. Semigroup Forum 3, 160–167 (1971). https://doi.org/10.1007/BF02572956

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  • DOI: https://doi.org/10.1007/BF02572956

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