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On regular semigroups whose idempotents form a semigroup: Addenda

Published online by Cambridge University Press:  17 April 2009

T. E. Hall
T. E. Hall
Affiliation:
Department of Mathematics, University of Stirling, Scotland.
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The results in the first parts of Theorems 2 and 3 of the paper in the title (see [2]) have been previously obtained by B.M. Schein in Theorem 1.12, page 299 [4], and in Proposition 1.13 (combined with the last paragraph of page 300) [4], respectively. To deduce the first part of Theorem 2 [2] from Theorem 1.12 [4] one merely uses the fact that a binary relation R on a set X satisfies RR−1R ⊆ R if if and only if it satisfies: R{x) ∩ R(y) ≠ □ implies R(x) = R(y), for any x, y ∈ X (see Proposition 9, page 132 [3]).

Type
Addendum
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 2 , October 1970 , pp. 287 - 288
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Clifford, A.M. and Preston, G.B., The algebraic theory of semigroups (Math. Surveys 7(I), Amer. Math. Soc., Providence, Rhode Island, 1961).Google Scholar
[2]Hall, T.E., “On regular semigroups whose idempotents form a subsemigroup”, Bull. Austral. Math. Soc. 1 (1969), 195–208.CrossRefGoogle Scholar
[3]Riguet, J., “Relations binaires, fermetures, correspondances de Galois”, Bull. Soc. Math. France 76 (1948), 114–155.CrossRefGoogle Scholar
[4]Šaĭn, B.M., [= B.M. Schein], “On the theory of generalized groups and generalized heaps” (Russian), Theory of semigroups and appl. I (Russian), 286–324, (Izdat. Saratov. Univ., Saratov, 1965).Google Scholar