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[A partial groupoid (half-magma) is a set S equipped with a (single-valued) partial binary operation, as in Bruck's Survey of Binary Systems.]

This question may be nonsensical, given that the duality between Cayley graphs and groups is deeply related to the mutual complete-ness of each structure. If that is the case, perhaps what I am really asking for is a common diagrammatic representation of partial groupoids. Here is a crappy mock-up of what I'd expect from diagrams of half-magmas in the latter case.

If Cayley graphs encode the abstract structure of group, is there a similar type of graphs that encode the abstract structure of a partial groupoid? If there is not, is there some other commonly used diagrammatic representation used for partial groupoids?

Thanks!

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  • $\begingroup$ Hi there. Please add your question to the post (not just the title) and make it as specific as possible. $\endgroup$
    – Martin Brandenburg
    Commented Dec 2, 2023 at 4:08
  • $\begingroup$ hopefully my last paragraph is more to the point now $\endgroup$
    – shea
    Commented Dec 2, 2023 at 4:29
  • $\begingroup$ There is so little structure in the concept of a partial groupoid that I cannot imagine anything at all useful to say. All it consists of is a subset $X \subset S \times S$ and a function $f : X \to S$. Think of the case where $S=\mathbb R$, and think of any silly subset $X \subset \mathbb R^2$ and any silly function $f : X \to \mathbb R$. What is there to be said in that level of generality? You can certainly define it's graph $\{(x,y,z) \mid z=f(x,y)\}$, but then what? $\endgroup$
    – Lee Mosher
    Commented Dec 2, 2023 at 15:26
  • $\begingroup$ as seen in Bruck's book, there is a surprising amount of technology that we may define on partial groupoids! sub-structures, extensions, chains, morphisms, and (free) generators all make an appearance in the first 8 pages (the rest of the book moves on to higher level structures). $\endgroup$
    – shea
    Commented Dec 3, 2023 at 0:47

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