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One variable equations over semigroups

Published online by Cambridge University Press:  17 April 2009

Frank Levin
Frank Levin
Affiliation:
Rutgers, The State University, New Brunswick, New Jersey, USA.
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Abstract

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An analogue of the theorem on the existence of a primitive element for separable extensions of fields is presented for semigroups. This has two immediate consequences.

(i) A semigroup is algebraically closed with respect to equations in several variables if and only if it is closed with respect to equations in a single variable.

(ii) Any countable semigroup C is embedded in a two-generator semigroup, one of whose generators is in C.

Further, a proof is given that any free product of a semigroup of order one with one of order two is SQ–universal, that is, its factor semigroups embed all countable semigroups. The proofs are adaptations of one used by Trevor Evans, Proc. Amer. Math. Soc. 3 (1952), 614–620. to show that a free product of two infinite cyclic semigroups is SQ–universal.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 2 , April 1970 , pp. 247 - 252
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Clifford, A.M. and Preston, G.B., The algebraic theory of semigroups, Vol. II (Math. Surveys 7 (II), Amer. Math. Soc., Providence, Rhode Island, 1967).Google Scholar
[2]Evans, Trevor, “Embedding theorems for multiplicative systems and protective geometries”, Proc. Amer. Math. Soc. 3 (1952), 614–620.CrossRefGoogle Scholar
[3]Levin, Frank, “One variable equations over groups”, Arch. Math. 15 (1964), 179–188.CrossRefGoogle Scholar