What is the name for inverse semigroups that have "unique factorisation of products"?

Question

This is a terminology/definition question. It does not require the usual level of context.

The Question:

What is the name for inverse semigroups that have "unique factorisation of products"?

The Details:

Definition: By "unique factorisation of products", I mean for all $a_1,\dots, a_r$ and all $b_1,\dots, b_s$ in the inverse semigroup $S$, if $$a_1\dots a_r=b_1\dots b_s,$$ then $r=s$ and $a_i=b_i$ for all $i$.

Thoughts:

If I recall correctly, not all inverse semigroups have this property, but I could be wrong.

Context:

My undergraduate dissertation was on inverse semigroups. That was a long time ago now.

There seems to be a different use of the word "factorisation" in the literature I have seen for inverse semigroups so far.

Answer 1

As you’ve stated it, only the empty semi group has this property. For suppose $x \in S$. Let $r = 2, s = 1, a_1 = a_2 = x, b_1 = xx$. Then $a_1 a_2 = b_1$ but $2 \neq 1$.

Answer 2

This is a lengthy comment/discussion, more than an answer to the original question.

It seems you're trying to reproduce the UFD property from rings to semigroups. As others pointed out, as stated, you either end up with empty set semigroup (some people don't consider that a semigroup), or the one element semigroup (dropping the requirement that $r$ and $s$ can be distinct).

If we think of prime elements of a ring as generators of the multiplicative structure, the analogue of uniqueness of writing an element $s\in S$ in terms of generators $s = x_1\cdot ...\cdot x_n$ might be interpreted as $S$ being the free monoid with variables $\{x_i\}_{i\in I}$.

If we want the uniqueness be up to rearrangement, we obtain the free commutative monoid with variables $\{x_i\}_{i\in I}$.

To further mimick UFD property we might want to introduce units as well, in which case we obtain $S = G\times T$ where $G$ is a group and $T$ is a free (commutative?) monoid, where I purposefully use direct product to avoid worrying about relations between elements of $G$ and $T$.

NOW, it would be interesting to see which inverse semigroups have unique factorization property in the sense that they're isomorphic to $G\times T$ for a group $G$ and free (commutative?) monoid $T$.

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Why monoids and not semigroups? It seems to me that the problem is easier to work with assuming that $S$ is a monoid. Moreover, this is the case where such semigroups arise the most, as the multiplicative monoid of a ring.