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In this old paper D. B. McAlister has introduced another 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 $a, b\in S.$

Both $\operatorname{ker}\theta=\{(a,b)\in S\times S : \theta(a)=\theta(b)\}$ and the image $\operatorname{im}\theta$ may not be closed under multiplication, in general. Therefore the quotient $S/\operatorname{ker}\theta$ is may not be an inverse semigroup, and $\operatorname{im}\theta$ need not to be a inverse subsemigroup of $T.$ 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, What are the least properties, except the multiplicativity, that we should expect from v-prehomomorphisms in order to formulate the first isomorphism theorem for inverse semigroups together with them?

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    $\begingroup$ According to Mark Lawson's book prehomomorphisms correspond to morphisms of ordered groupoids. This is a more complicated setting and so you can't I think hope to formulate such a nice theorem unless the prehomomorphism is injective on idempotents. Ordered groupoids are essentially etale groupoids where you give posets the alexandrov topology and the morphisms are continuous functors. You might be ok of the prehomomorphism is an open map or a local homeomorphism which means surjective (resp bijective) on principal lower sets $\endgroup$
    – Benjamin Steinberg
    Commented Jul 21, 2021 at 19:02
  • $\begingroup$ @BenjaminSteinberg: I didn't know any reference to prehomomorphisms other than the linked paper. Thank you for mentioning Lawson's book. I will look at it. Also, if you like to turn your comment into an answer, I am happy to accept it. $\endgroup$
    – Bumblebee
    Commented Jul 21, 2021 at 21:01
  • $\begingroup$ I'm not sure the comment is really an answer but the book will tell you a lot about prehomomorphisms. $\endgroup$
    – Benjamin Steinberg
    Commented Jul 21, 2021 at 21:07

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