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The Embedding Theorems of Malcev and Lambek

Published online by Cambridge University Press:  20 November 2018

George C. Bush
George C. Bush*
Affiliation:
Queen s University
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A given semigroup is said to be embeddable in a group if there exists a group which contains a subsemigroup isomorphic to .

It can easily be proved that cancellation is a necessary condition for embeddability. It can also be shown that we can adjoin an identity to a semigroup without identity in such a way that the new semigroup is embeddable if and only if the original was embeddable. Therefore, we can, without loss of generality, restrict our attention to cancellation semigroups with identity, whenever this is convenient.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 15 , 1963 , pp. 49 - 58
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Bush, G. C., On embedding a semigroup in a group, Ph.D. thesis, Queen's University, Kingston, Ontario (1961).Google Scholar
2. Lambek, J., The immersibility of a semigroup into a group, Can. J. Math., 3 (1951), 34–43.Google Scholar
3. Malcev, A., Über die Einbettung von assoziativen Systemen in Gruppen, Mat. Sbornik, 6 (48) (1939), 331–336.Google Scholar
4 Malcev, A., Uber die Einbettung von assoziativen Systemen in Gruppen, II, Mat. Sbornik, 8 (50) (1940), 251–264.Google Scholar
5. Tamari, D., Monoïdes préordonnés et chaînes de Malcev, Bull. Soc. Math. France, 82 (1954), 53–96.Google Scholar