Extending semigroup theory? In an earlier question I proposed a definition of associativity for ternary relations generalizing the usual notion for composition laws. I'm still not sure whether this definition makes sense, but if so this leads to some interesting algebraic questions. In particular, what parts of "semigroup theory" can be extended to that setting, if any?

Consider such a relation $\cal{R}$ over $X$. Similar to semigroup theory, we can give the following definitions:


a set $S \subseteq X$ is a "substructure" if $a,b \in S$ and $\cal{R}(a,b,c)$ holds imply that $c \in S$;
a set $L \subseteq X$ is a "left ideal" if $a \in L, b \in X$ and $\cal{R}(a,b,c)$ holds imply that $c \in L$;
a set $R \subseteq X$ is a "right ideal" if $a \in X, b \in R$ and $\cal{R}(a,b,c)$ holds imply that $c \in R$;
given a substructure $S$, we can define an order relation $\leq_S$ such that for any $a,a' \in X$, $a \leq_S a'$ iff for every $b \in S,c \in X$, $\cal{R}(a,b,c)$ implies $\cal{R}(a',b,c)$. We can also define an equivalence relation $\equiv_S$ as $\leq_S \cap \geq_S$.


This leads to some natural questions: (i) what are the conditions on $S$ such that $\cal{R} / \equiv_S$ behaves as a substructure? (ii) how do chains of ideals behave with respect to inclusion? (iii) are there some possible generalizations of the isomorphism theorems? (iv) are there some connections with automata and possible extensions of Krohn-Rhodes theory?

This is probably pie in the sky though, as I don't even have any concrete example of such a structure...
The structure you suggest is known as a 'semihypergroup', a notion introduced by F. Marty in 1934. According to Google, some of your questions have already been attacked by a diverse community of mathematicians involving Greeks, Iranians and Thai people. The most challenging question seems point (iv) but I'm unsure about the outcome, though a positive result in that directions would probably have far-reaching consequences -- possibly in relation with Rhodes' interest in the separation of complexity classes.