Inverse Semigroup - Homomorphisms and Representations of Inverse Semigroups

Homomorphisms and Representations of Inverse Semigroups

A homomorphism (or morphism) of inverse semigroups is defined in exactly the same way as for any other semigroup: for inverse semigroups S and T, a function θ from S to T is a morphism if (sθ)(tθ) = (st)θ, for all s,t in S. The definition of a morphism of inverse semigroups could be augmented by including the condition (sθ)−1 = s−1θ, however, there is no need to do so, since this property follows from the above definition, via the following theorem:

Theorem. The homomorphic image of an inverse semigroup is an inverse semigroup; the inverse of an element is always mapped to the inverse of the image of that element.

One of the earliest results proved about inverse semigroups was the Wagner-Preston Theorem, which is an analogue of Cayley's Theorem for groups:

Wagner-Preston Theorem. If S is an inverse semigroup, then the function φ from S to, given by

dom (aφ) = Sa−1 and x(aφ) = xa

is a faithful representation of S.

Thus, any inverse semigroup can be embedded in a symmetric inverse semigroup.