Let $H$ denote a subsemigroup of a semigroup $G$.
I'm interested in the following property: $$\forall g\in G\exists h\in H:gh\in H.$$ This property is weaker than the property that $H$ is an ideal in $G$. Does the property have a name? Are there related properties?
Looking forward to your comments.
This property does not have a name. It is true for every subsemigroup $H$ containing a right ideal of $G$. But it is also true in many other cases. For example if $G$ is a linearly ordered group (no non-trivial right ideals), and $H$ a positive cone of $G$ (all elements that are $\ge 1$), then it satisfies your property.
It is not clear from your question how related the "related" properties should be. If you say something about your motivation, it would be easier to answer.
Update No, it has nothing to do with any of the (several different) definitions of a normal subsemigroup because normal subgroups of groups are normal subsemigroups according to any of these definitions. Normal (or any other) subgroups of groups do not satisfy your property. If $G$ is a free semigroup, then it is linearly ordered and has many right ideals, so it has many subsemigroups $H$ satisfying your property. One can characterize them all by saying that $H$ satisfies the property if and only if every word $w$ is a left quotient of two words from $H$, i.e. $pw=q$ where $p,q\in H$.
