In this old paper D. B. McAlister has introduced a very interesting class of morphisms for inverse semigroups, called v-prehomomorphisms. For such a morphism $\theta : S \to T,$ instead of preserving the multiplication, he require
- $\theta(ab)\le \theta(a)\theta(b)$
- $\theta(a^{\star})=\theta(a)^{\star}$
for all $a, b\in S.$ Previously, here I asked a question about these maps.
Since the image $\operatorname{im}\theta$ may not be closed under multiplication, it need not to be a inverse subsemigroup of $T.$ Also, $\operatorname{ker}\theta=\{(a,b)\in S\times S : \theta(a)=\theta(b)\}$ is not compatibility with multiplication, and hence the quotient $S/\operatorname{ker}\theta$ is not an inverse semigroup, in general. However, there is still a bijection $S/\operatorname{ker}\theta \to \operatorname{im}\theta$ of sets such that $[a]\mapsto \theta(a).$
Now my question is, Is there any way around these deficiencies to formulate the first isomorphism theorem for inverse semigroups together with v-prehomomorphisms?