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Semigroup Forum

, Volume 12, Issue 1, pp 183–184 | Cite as

Note on the orderability of idempotent semigroups

  • Thérèse Merlier
Short Note

Abstract

In [3] , we gave a condition for the orderability of finite idempotent semigroups. Recently, in [5], T. Saitô gives a condition for the orderability of idempotent semigroups in the general case. As Corollary ( 4–13 of [5] ), he obtains an idempotent semigroup S is orderable if and only if every finite subsemigroup of S is orderable. The purpose of this note is to give a direct proof of this result.

Keywords

Direct Proof Idempotent Semigroup 
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References

  1. 1.
    Green, J.A. and D. Rees,On semigroups in which xr = x, Proc. Camb. Phil. Soc. 48 (1952), 35–40.Google Scholar
  2. 2.
    Kreisel, G. et J.L. Krivine,Eléments de Logique Mathématique,Théorie des Modèles, Dunod Paris 1967.Google Scholar
  3. 3.
    Merlier, Th.Sur les σ-bandes finies et les demi-groupes totalement ordonnés d-simples, Semigroup Forum 4 (1972), 124–149.Google Scholar
  4. 4.
    Neumann, B.H.,An embedding theorem for algebraic systems Proc. London. Math. Soc. 4 (1954), 138–153.Google Scholar
  5. 5.
    Saitô, T.The orderability of idempotent semigroups. Semigroup Forum 7 (1974), 264–285. ordered algebraic systems, Pergamon Press (1963).Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1976

Authors and Affiliations

  • Thérèse Merlier
    • 1
  1. 1.PARIS- Université Pierre et Marie CurieFontenay aux RosesFrance

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