Suppose I define an algebra with 3 or more operations, perhaps using universal algebra. Would it be meaningful to talk about the representation theory of this algebra?
In particular, I am interested in relation algebras, these have 3 or more binary operations and at least 2 unary operations (negation and complement) obeying rather complicated distributive laws.
Here is another example with $3$ operations. We are studying post-Lie algebras, which have three operations on a given vector space, namely two Lie brackets $[x,y]$ and $\{x,y\}$ and another bilinear product $x\cdot y$. All three operations are interwined in a certain way. They arise, among other things, in simply transitive actions of a Lie group $G$ on another Lie group $N$ by affine transformations. One may associate to it Post-Lie algebra modules.