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.