Every finite semigroup is embeddable in a finite relatively free semigroup |
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Authors: | George M Bergman |
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Institution: | Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA |
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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 ξd=ξ2d 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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Keywords: | Primary: 20M05 secondary: 20M07 20M30 |
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