Infinite semigroups whose non-trivial homomorphs are all isomorphic
Bruce Jensen · Pacific Journal of Mathematics · 1969
An infinite semigroup S such that every nontrivial homomorph of it is isomorphic to S is called an HI semigroup.Every commutative HI semigroup is a group and thus it is isomorphic to the group Z(p)°° 9 for some prime P.An infinite Brandt semigroup is HI if and only if it has a trivial structure group.An inverse HI semigroup containing a primitive idempotent is either Brandt or else it is isomorphic to a trasfinite chain of extensions of a Brandt semigroup K by isomorphic copies of K (where K has the trivial group as its structure group).Necessary and sufficient conditions are given for a semigroup of the latter type to yield an HI semigroup and an example is constructed.