I have in mind a structure similar to a group, but different in that the binary operation is undefined for some pairs of elements. For lack of a better term, call $G$ a restricted group if there is a binary operation $\star:S\subset G \times G \to G$ such that
(1) There is an identity element $e$ such that for all $g \in G$, we have $e \star g = g \star e = g$.
(2) For each $g \in G$ there exists an element $g^{-1} \in G$ such that $g \star g^{-1} = g^{-1} \star g = e$.
(3) For all $f,g,h \in G$ such that $f \star g$, $(f \star g) \star h$, $g \star h$ and $f\star(g \star h)$ are all defined according to the binary operator, then $$(f \star g) \star h = f\star(g \star h).$$
I imagine this structure must have been studied before and given some name. For instance, I could imagine in robotics there are some physically restricted domains. My particular motivation for asking about this comes from bandaged rubik's cube puzzles, where pieces are glued together.
This is called a groupoid. A naturally occurring example is the fundamental groupoid which has a binary operation when concatenation is possible.