What type of property for subsemigroups or submonoids is this? Consider the set on nonzero-rational numbers $\mathbb{Q}^*= \mathbb{Q}-\{0\}$ as a subsemigroup of the nonzero real numbers $\mathbb{R}^*=\mathbb{R}-\{0\},$ where the semi-group operation $\cdot$ is multiplication. As a subsemigroup, $\mathbb{Q}^*$ has the additional property that:
$$\forall x\in \mathbb{R}^* \forall y\in \mathbb{Q}^*(x\cdot y\in \mathbb{Q}^*\Rightarrow x\in \mathbb{Q}^*). \tag{1}$$
Can you give me the name for subsemigroups obeying property (1) please? Also, consider the same question, but for monoids, not semigroups. What is the property in that case?
(edit: previously I had asked a similar question, replacing “semigroup” with “group”. The answer in that case is all subgroups obey property (1).)
 A: A subset $P$ of a semigroup is right unitary if for all $p \in P$ and $s \in S$, $sp \in P$ implies $s \in P$.
It is left unitary if for all $p \in P$ and $s \in S$, $ps \in P$ implies $s \in P$. It is unitary if it is both left and right unitary. This definition applies of course to subsemigroups.
If you are looking for examples, the class of $E$-unitary semigroups (semigroups in which the set of idempotents form a unitary subsemigroup) has been extensively studied, both in the regular case and in the general case [1].
EDIT. Another famous example: a submonoid of a free monoid is free if and only if it is unitary.
Also note that there is a similar definition for categories. A subcategory $N$ of a category $C$ is said to be unitary if for all $x, y \in {\text Mor}(C)$,

*

*if $xy, x \in {\text Mor}(N)$, then $y \in {\text Mor}(N)$ and

*if $xy, y \in {\text Mor}(N)$, then $xy \in {\text Mor}(N)$.

[1] J. Almeida, J.-É. Pin and P. Weil, Semigroups whose idempotents form a subsemigroup, Math. Proc. Cambridge Phil. Soc. 111 (1992), 241-253.
