# Normal sub-semi-group

of a semi-group $S$
A sub-semi-group $H$ satisfying the following condition: For any $x,y\in S^1$ (for the notation $S^1$ see Normal complex) such that $xy\in S$ and for any $h\in H$ the relations $xhy\in H$ and $xy\in H$ are equivalent. A subset of $S$ is a normal sub-semi-group if and only if it is the complete inverse image of the unit element under some homomorphism of $S$ onto a semi-group with unit element.