Say $h$ is related to $g$ in some way, denote this by $h:R:g$. Say we also have some kind of operation that we can perform on both $h$ and $g$. Denote this operation by $*$. The relation, $R$, is such that:
$$hRg \implies h*x :R: g*x$$
For example, if $h, g, x \in \mathbb{Z}$, the relation $~\geq~$ with the operation $[\text{multiplication}]$ does not have this property, because one could multiply by $-1$. It would if $h, g, x \in \mathbb{Z}^+$. Equality has this property with any operation (as does any equivalence relation, I think).
Is there a name for this property?
I understand this is somewhat abstract, and I may be ignorant of a whole slew of terminology. To put this in some context, this question came from my having to prove to myself that, if $H$ is a normal subgroup of $G$, then $g*H*g^{-1} = H ~~\forall g \in G$ (where $*$ in this case is the group operation). We certainly know $g*H*g^{-1} \subseteq H$. Group-operating by $g$ on the right and $g^{-1}$ on the left, assuming $(\subseteq, *)$ has this property, would show $H \subseteq g^{-1}*H*g = g*H*g^{-1}$.
(Indeed, $(\subseteq, *)$ has this property)
