Separable semi-group

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) Zbl 0111.03403