# Every finite semigroup is embeddable in a finite relatively free semigroup

- Authors
- Journal
- Journal of Pure and Applied Algebra 0022-4049
- Publisher
- Elsevier
- Publication Date
- Volume
- 186
- Issue
- 1
- Identifiers
- DOI: 10.1016/s0022-4049(03)00126-9

## Abstract

Abstract The title result is proved by a Murskii-type embedding. Results on some related questions are also obtained. For instance, it is shown that every finitely generated semigroup satisfying an identity ξ d = ξ 2 d is embeddable in a relatively free semigroup satisfying such an identity, generally with a larger d; but that an uncountable semigroup may satisfy such an identity without being embeddable in any relatively free semigroup. It follows from known results that every finite group is embeddable in a finite relatively free group. It is deduced from this and the proof of the title result that a finite monoid S is embeddable by a monoid homomorphism in a finite (or arbitrary) relatively free monoid if and only if its group of invertible elements is either { e} or all of S.

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