Let $S$ be a semigroup. An ideal (of $S$) is a subset $I$ of $S$ such that $SI$ and $IS$ are both contained in $I$. The non-empty ideals constitute a subsemigroup, $\mathfrak I(S)$, of the power semigroup of $S$ under the operation of setwise multiplication induced by $S$ on its parts; and for some reasons, one may want to ensure that the finitely generated ideals constitute a subsemigroup of $\mathfrak I(S)$. However, this is not going to be the case if a finitely generated ideal is taken to be a set of the form $X \cup XS \cup SX \cup SXS$ for some finite $X \subseteq S$ (which looks like the most obvious/natural thing to do). It goes the same with principal ideals, which are commonly defined as monogenic ideals, that is, sets of the form $\{a\} \cup aS \cup Sa \cup SaS$ with $a \in S$.
On the other hand, letting a finitely generated ideal be an ideal $I$ such that $I = X \cup XS = Y \cup SY$ for some finite sets $X, Y \subseteq S$ would turn the set of all finitely generated ideals into a subsemigroup $\mathfrak I_{\rm fin}(S)$ of $\mathfrak I(S)$ (conditional is mandatory here). Under this "alternative definition", a monogenic ideal would then be an ideal $I$ such that $I = \{a\} \cup aS = \{b\} \cup Sb$ for some $a, b \in S$, and the monogenic ideals would form a subsemigroup of $\mathfrak I_{\rm fin}(S)$. In a way, this makes a lot of sense, as we are asking that $I$ is finitely generated (resp., principal) both as a left and as a right ideal; and these left and right conditions are, in principle, independent from each other.
Question(s). Have these "alternative notions" (of finitely generated ideals and principal ideals) been considered before? If yes, under which names?
Note that the classical notions and the alternative ones coincide when $S$ is a commutative semigroup (or, more generally, a duo semigroup). In addition, any finitely generated (resp., principal) ideal in the alternative sense is a finitely generated (resp., principal) ideal in the classical sense.
