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Separable semi-group

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A semi-group in which for elements , implies that . If a semi-group has a partition into sub-semi-groups that satisfy the cancellation law, then is separable. For commutative semi-groups the converse holds; moreover, any commutative separable semi-group can be expanded into a band of semi-groups (trivially into a semi-lattice) with the cancellation law. A commutative semi-group is separable if and only if it can be imbedded in a Clifford semi-group. A periodic semi-group is separable if and only if it is a Clifford semi-group. A commutative semi-group is separable if and only if its characters separate the elements of .

References

[1] A.H. Clifford, G.B. Preston, "The algebraic theory of semigroups" , 1 , Amer. Math. Soc. (1961)
How to Cite This Entry:
Separable semi-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Separable_semi-group&oldid=31737
This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article