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.

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    $\begingroup$ en.wikipedia.org/wiki/Groupoid#Algebraic $\endgroup$ – Ayman Hourieh Jun 9 '18 at 20:06
  • $\begingroup$ In your (3), the most common definitions use " if $fg$ and $(fg)h$ are defined, then so are $gh$ and $f(gh)$ and we have $(fg)h=f(gh)$"; at least that's what happens for groupoids $\endgroup$ – Max Jun 9 '18 at 20:20
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    $\begingroup$ How about "incomplete group"? $\endgroup$ – Somos Jun 9 '18 at 20:57

This is called a groupoid. A naturally occurring example is the fundamental groupoid which has a binary operation when concatenation is possible.


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