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Let $S$ be a semigroup and $\widehat{S}$ be its unitization, i.e., the monoid obtained from $S$ by adjoining an identity element if necessary (so that $\widehat{S} = S$ when $S$ is already a monoid).

We say that $a$ divides (or is a divisor of) $b$ in $S$, and we write $a \mid_S b$, if $a, b \in S$ and $b = uav$ for some $u, v \in \widehat{S}$. A divisor-closed subsemigroup of $S$ is then a subsemigroup $T$ of $S$ such that, if $a \mid_S b$ and $b \in T$, then $a \in T$. E.g., the empty semigroup and $S$ are both divisor-closed subsemigroups of $S$.

Given a set $X \subseteq S$, the intersection of all divisor-closed subsemigroups of $S$ containing $X$ is itself a divisor-closed subsemigroup of $S$ (containing $X$), hereinafter denoted by $[\![X]\!]_S$. This notion is quite useful when it comes to studying the arithmetic of semigroups, partly because it is often possible to show that certain properties hold "locally" (i.e., for the divisor-closed subsemigroups of the form $[\![x]\!]_S$ with $x \in S$) if and only if they hold globally. For instance, the semigroup $S$ is atomic (i.e., every non-unit of $\widehat{S}$ factors as a product of atoms) iff $[\![x ]\!]_S$ is atomic for every non-unit $x \in S$. (Here, an atom of a semigroup $N$ is a non-unit of its unitization $\widehat{N}$ that does not factor as a product of two non-units in $\widehat{N}$, and I'm writing $[\![x ]\!]_S$ in place of $[\![\{x\} ]\!]_S$.)

When $S$ is, say, commutative, $[\![X]\!]_S$ has a neat description — it is the subsemigroup (of $S$) generated by the divisors of the elements of the subsemigroup generated by $X$. Things are not so smooth in general. However, it is still possible to describe $[\![X]\!]_S$ as the union of the terms of the sequence $D_0(X)$, $D_1(X)$, $\ldots$ of subsets of $S$ defined by taking $D_0(X) := X$ and letting $D_n(X)$ be, for each $n \in \mathbb N^+$, the subsemigroup generated by the divisors of the elements in $D_{n-1}(X)$. In some situations, this is more than enough to do something, which leads me to the following:

Question. Do you have a reference for the "iterative description" of the subsemigroup $[\![X]\!]_S$ outlined in the above?

It is only recently that I've seen divisor-closed subsemigroups being used in the non-commutative setting (I myself have used them more than once), but that's probably ignorance from my side. It is highly likely that divisor-closedness is referred to in a different way in the larger literature on semigroups (in my field, the term is well established). Clarifying this aspect is also part of the motivation for the question.

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  • $\begingroup$ Note an answer, just a comment: Whenever there is a closure operator in play, you have a top-down construction (the intersection you mentioned) and a bottom-up construction (closing, then closing again, etc...). So, if you are satisfied with showing that there is a natural closure operator, you could just refer to the literature on closure operators for the "iterative description" you want. (In your case it stabilizes in countable many steps because the operator is finitary.) $\endgroup$
    – Pace Nielsen
    Commented Feb 18 at 16:08
  • $\begingroup$ @PaceNielsen Do you have in mind any candidate for the closure operator alluded to in your comment? I'm not sure whether I would be satisfied with this approach, but I'm a big fan of high tech. I guess that much will ultimately depend on the details. $\endgroup$
    – Salvo Tringali
    Commented Feb 18 at 17:21
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    $\begingroup$ @PaceNielsen I had never realized that "the subgroup generated by" and "the set of divisors" can be viewed as finitary closure operators. That's cool! Let me denote the 1st operator by $\Pi$, the 2nd by $\Delta$, and the composition "$\Pi$ after $\Delta$" by $\Gamma$. I agree that, in the notation of the OP, $D_n(X)=\Gamma^{\circ n}(X)$ for all $n\in\mathbb N$, where $\Gamma^{\circ n}$ is the $n$th iterate of $\Gamma$. How does this show that $[\![X]\!]_S=\bigcup_n\Gamma^{\circ n}(X)$ from general principles? Not very familiar with closure operators and clearly missing something. Any refs? $\endgroup$
    – Salvo Tringali
    Commented Feb 19 at 5:40
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    $\begingroup$ It is not true that $\bigcap_{Y\in C(X)}Y = \bigcap_{X\subseteq Y\subseteq S}\Gamma(Y)$ because $\Gamma$ itself might not be a closure operator. (The composition of closure operators is not necessarily a closure operator.) The set on the left is $\Gamma$-closed (and hence both $\Pi$- and $\Delta$-closed), and hence equals $\bigcup_{n}\Gamma^{\circ n}(X)$ [since this is $\Gamma$-closed by the argument I gave, and clearly minimal]. $\endgroup$
    – Pace Nielsen
    Commented Feb 20 at 14:58
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    $\begingroup$ I believe "Exercise 6.3:4(ii)" would be an appropriate reference for the claim about the joint closure stabilizing at the $\omega$th stage. [Writing the semigroup operation multiplicatively, take the set $G$ in that exercise to consist of all pairs $(\{x,y\},xy)$ (forcing closure under being a semigroup) together with all pairs $(\{x\},d)$ whenever $d$ is a divisor of $x$ (forcing closure under divisors).] $\endgroup$
    – Pace Nielsen
    Commented Feb 20 at 17:14

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