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Free products amalgamating unitary subsemigroups

Published online by Cambridge University Press:  17 April 2009

G.B. Preston
G.B. Preston
Affiliation:
Mathematical Institute, University of St Andrews, St Andrews, Scotland. Department of Mathematics, Monash University, Clayton, Victoria.
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Abstract

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Let Si, i ∈ I, be a set of semigroups such that Si ∩ Sj = U, if i ≠ j, and such that U is a unitary subsemi-group of Si for each i in I. The semigroup amalgam [{Si | i ∈ I}; U] determined by this system is the partial groupoid G = USi in which a product of two elements is defined if and only if they both belong to the same Si and their product is then taken as their product in Si. In 1962, J.M. Howie showed that the amalgam G is embeddable in the free product of the Si, amalgamating U. To prove this result it suffices to find any semigroup in which G can be embedded. In this paper, by taking convenient representations of the Si, adapting a method recently (1975) used by T.E. Hall for inverse semigroups, we provide a short method of constructing such a semigroup.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 15 , Issue 1 , August 1976 , pp. 117 - 124
Copyright
Copyright © Australian Mathematical Society 1976

References

[1]Hall, T.E., “Free products with amalgamation of inverse semigroups”, J. Algebra 34 (1975), 375–385.CrossRefGoogle Scholar
[2]Howie, J.M., “Embedding theorems with amalgamations for semigroups”, Proc. London Math. Soc. (3) 12 (1962), 511–534.CrossRefGoogle Scholar
[3]Preston, G.B., “Free products amalgamating almost unitary subsemigroups”, submitted.Google Scholar