Let $S$ be a semigroup and $\widehat{S}$ be 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 $b$ in $S$, and 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 $x \in S$. (An atom of $S$ is a non-unit of $\widehat{S}$ that does not factor as a product of two non-units of $\widehat{S}$, and I'm writing $[\![x ]\!]_S$ in place of $[\![\{x\} ]\!]_S$.)

When $S$ is, say, commutative, $[\![X]\!]_S$ have a very neat description — it is the subsemigroup of $S$ generated by the set $\{y \in S: y \mid_S x^n, \text{ for some }n = 1, 2, \ldots\}$. Things are not so smooth in general. However, it is still possible to describe $[\![X]\!]_S$ as the union of the terms of a recursively defined sequence $D_0(X)$, $D_1(X)$, $\ldots,$ of subsets of $S$, where $D_0(X) := X$ and, for each $n \in \mathbb N^+$, $D_n(X)$ is the subsemigroup of $S$ generated by the set

$$ \{y \in S : y \mid_S x, \text{ for some }x \in D_{n-1}(X)\}. $$

In some situations, this is more than enough and leads us straight to the following:

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

To be honest, 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 question.