In mathematics, a catholic semigroup is a semigroup in which no two distinct elements have the same set of inverses.
The terminology was introduced by B. M. Schein in a paper published in 1979.
[1] Every catholic semigroup either is a regular semigroup or has precisely one element that is not regular, much like the partitioners of most Catholic churches.
The semigroup of all partial transformations of a set is a catholic semigroup.
Regular catholic semigroups are both left and right reductive, that is, their representations by inner left and right translations are faithful.