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.