Weaker version of inverse semigroup

Question

Is there a weaker version of inverse semigroup where the existence of reflexive inverse is dropped, only the uniqueness remains.

This was motivated by, and resembles the weaker version of divisible: cancellable -- the former requires both existence and uniqueness, while the latter only requires uniqueness.

More formally, a "weakly inverse semigroup" is a semigroup $S$ such that $\forall a, p, q \in S\ (apa=a \wedge pap=p \wedge aqa=a \wedge qaq=q \rightarrow p=q)$.

This concept will also be a weaker version of cancellative semigroup. (I think)

Is there such a concept being used anywhere? Or if not, is there any reason why not (such as having no interesting property, etc.).

Answer 1

According to your definition, a semigroup is weakly inverse if each of its elements has at most one inverse. Now, an element has at least one inverse if and only if it is regular. Thus a semigroup is weakly inverse if and only if every regular element has exactly one inverse.

This class of semigroups has been widely studied, at least in the finite case, under the name of block groups, because of the very peculiar structure of their regular $\cal D$-classes, represented in the picture below, where $G$ is a maximal group of the $\cal D$-class and a star indicates the presence of an idempotent in the corresponding $\cal H$-class:

$\hskip 100pt$

Block-groups admit several characterizations in terms of Green relations.

Theorem. Let M be a finite monoid. The following conditions are equivalent:

1. M is a block group;
2. for every regular $\cal D$-class $D$ of $M$, $D^0$ is a Brandt semigroup;
3. for all idempotent $e, f$ of $M$, $e \mathrel{\cal R} f$ implies $e=f$ and $e \mathrel{\cal L} f$ implies $e=f$;
4. for all idempotent $e, f$ of $M$, $efe=e$ implies $ef = e = fe$;
5. the submonoid generated by the idempotents of $M$ is $\cal J$-trivial.

Finite block-trivial monoids are closed under taking submonoids, morphic images (i.e. quotient monoids) and finite direct products and hence form a variety of finite monoids.

If $G$ is a group, then ${\cal P}(G)$, the monoid of subsets of $G$, under the operation defined by $$XY = \{xy \mid x \in X, y \in Y\}$$ is a block-group. A (difficult) theorem states that the variety of finite monoids PG generated by the monoids of the form ${\cal P}(G)$, where $G$ is a group, is equal to the variety BG of all finite block-groups. See [1] for a survey and further results.

[1] J.-É. Pin, PG = BG, a success story, in NATO Advanced Study Institute, Semigroups, Formal Languages and Groups, J. Fountain (ed.), Kluwer academic publishers (1995) 33-47.