Baer semi-group

A semi-group $S$ with an absorbing element 0 having the property that the right annihilator of any element is a principal right ideal on an idempotent element of $S$, and similarly the left annihilator of any element is a principal left ideal on an idempotent element of $S$.

Examples include the monoid of binary relations on a set $A$ under composition of relations, with the empty relation as zero element.

A Baer semigroup with involution is termed a Foulis semigroup.

References

[1]	T.S. Blyth Lattices and Ordered Algebraic Structures Springer (2006) ISBN 184628127X