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While exploring combinations of posets, I've come across a ternary relation $R$ on a 5-set (of posets) that looks a little like the operation for some group, though it's clearly not $C_5$... not least because $R$ isn't even univalent. I gather that some authors would describe $R$ as a multivalued function.

Here's what $R$ looks like, displayed as though it were a group operator: $$ \begin{array}{c|ccccc} & e & a & b & c & d \\\hline e & e & a & b & c & d \\ a & a & e & c & \mbox{$b$ or $d$} & c \\ b & b & c & e & \mbox{$a$ or $d$} & c \\ c & c & \mbox{$b$ or $d$} & \mbox{$a$ or $d$} & \mbox{$e$ or $c$} & \mbox{$a$ or $b$} \\ d & d & c & c & \mbox{$a$ or $b$} & e \\ \end{array} $$

So I have two questions: $$ \begin{array}{rl} 1. & \mbox{Are these operation-ish beasts a thing?} \\ 2. & \mbox{And for this particular $R$, does anybody recognize it from somewhere else?} \end{array} $$

UPDATE

I now have the answer to question 1. Beasts like $R$ have indeed been studied, and they are known in general as hyperstructures.

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    $\begingroup$ I've come across a ternary relation $R$ on a $5$-set --- Book, paper, lecture notes, blog post, ...? Knowing which might help others. In my case, I wonder if perhaps this was an exercise in which one is to choose specific values for each of the multi-valued entries to obtain an operation having a certain specified property. $\endgroup$ Commented Jan 13, 2020 at 15:17
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    $\begingroup$ I'm a little surprised by the $(c,d)$ vs. $(d,c)$ entries - everything else is symmetric, but there we have "$a,b$, or $c$" vs. "$a$ or $b$." Is that correct? $\endgroup$ Commented Jan 13, 2020 at 15:34
  • $\begingroup$ @DaveL.Renfro, I'm working a research problem in order theory, not asking for help with a homework problem. I hope that my edits to the question provide enough of the kind of context that you suggest may be useful. $\endgroup$ Commented Jan 13, 2020 at 15:38
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    $\begingroup$ The google searches "multigroup" + "multivalued" AND "group" + "multivalued operation" AND "multivalued binary operation" might turn up something. $\endgroup$ Commented Jan 13, 2020 at 15:49
  • $\begingroup$ @NoahSchweber, thanks for your question. It turns out I had done a poor job of fact-checking my table. I don't know why the asymmetry in the version I'd originally posted didn't catch my eye. In any event, I've fixed that mistake and I think the table is now correct. $\endgroup$ Commented Jan 13, 2020 at 15:52

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