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My question is closely related to this one, but the different choice of terminology might justify posting it.

One may define a category to be a graph with an associative composition of arrows and identity arrows (with respect to the composition).

Every nonassociative small magma (e.g nonassociative algebra) may be considered as the set of arrows of a graph with a one vertex, and hence it can be seen as a graph with a nonassociative composition. Also, one may construct 'artificial' graphs with more than one vertex and a nonassociative composition. But, what I am looking for is

Are there known 'natural' examples of graphs with multiple vertices and a nonassociative composition. Do such objects have a particular name? Are particular instances of such objects well-studied?

By 'natural', I am asking for such graphs not to be only constructed for the sole purpose of answering this question, but to arise from the study of more familiar objects.

References would be very appreciated.

Edit: I am not interested in an associative-like compositions, particularly compositions that are associative up to isomorphisms, as for $\infty$-categories. I am looking for something a bit more wild compared to categories, something as different from categories as Lie algebras (or BCK algebras) are different from associative algebras.

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user337830
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I honestly don't really "get" (motivation-wise) any of the examples, but if you have any hope of finding something interesting, I think you should try Plots and Their Applications - Part I: Foundations Section 5 "Some appealing examples" and also consider the references in this article, e.g. those listed in Section 2 "Background and philosophy".

A vast generalization of the notion of a category called "plot" is considered in this article, but in Section 2 many specialized notions previously considered are listed too.

I'd think you'd also find something useful in Lutz Schröder's PhD thesis "Composition graphs" (1999), but unfortunately (probably for you, not for me), it's in German. Sadly (in general) I can't seem to get a hold of this thesis.

Stefan Perko
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