Let $C$ be a semigroup (or analogously a category). A family $A ⊆ C$ is called
- *subsemigroup* if $a, b ∈ A \implies ab ∈ A$,
- *left ideal* if $b ∈ A \implies ab ∈ A$ (analogously we define right ideal and two-sided ideal).
Question: Are there names for the following properties?
- $ab ∈ A \implies b ∈ A$ (think of $A$ consisting of injective maps $X \to X$ and $S$ consisting of all maps; in general the family of all monomorphisms in a category have the property),
- $ab ∈ A ∧ a ∈ A \implies b ∈ A$ (think of $A$ consisting of distance preserving maps $X \to X$ on a metric space and $S$ consisting of all maps)?
Added: When $C$ is a is trivial preoder on a set $X$ (i.e. every two elements are comparable) viewed as a category, then $A ⊆ C$ corresponds to a binary relation on $X$, and the property 2. corresponds to the notion of Euclidean relation (https://en.wikipedia.org/wiki/Euclidean_relation).
According to Dubreil [2] (see also Clifford and Preston [1, p. 55]), a subset satisfying (2) is called left unitary.
For (1), recall that the preorder Green relation $\leqslant_\mathcal{L}$ is defined by $x \leqslant_\mathcal{L} b$ (read "$x$ is $\cal L$-below $b$") if and only if there exists $c \in C \cup \{1\}$ such that $x = cb$. Thus (1) means that $A$ is a $\leqslant_\mathcal{L}$-upper set.
[1] A. H. Clifford and G. B. Preston, The algebraic theory of semigroups. Vol. II. Mathematical Surveys, No. 7. American Mathematical Society, Providence, RI, 1967. xv + 350 pp.
[2] P. Dubreil, Contribution à la théorie des demi-groupes. Mém. Acad. Sci. Inst. France (2) 63 (1941), no. 3, 52 pp