Catholic semigroup

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. The semigroup of all partial transformations of a set is a catholic semigroup. It follows that every semigroup is embeddable in a catholic semigroup. But the full transformation semigroup on a set is not catholic unless the set is a singleton set. Regular catholic semigroups are both left and right reductive, that is, their representations by inner left and right translations are faithful. A regular semigroup is both catholic and orthodox if and only if the semigroup is an inverse semigroup.

See also

References
  1. Proceedings of the Conference in honour of A.H. Clifford. New Orleans. 1979. pp. 207–214.
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