# Separable semi-group

From Encyclopedia of Mathematics

A semi-group in which $x^2=xy=y^2$ for elements $x$, $y$ implies that $x=y$. If a semi-group $S$ has a partition into sub-semi-groups that satisfy the cancellation law, then $S$ 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 $S$ is separable if and only if its characters separate the elements of $S$.

#### 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=35783

This article was adapted from an original article by L.N. Shevrin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article