Isomorphic two structures but different type.

Question

If I have structure $(S, \cdot)$, where $\cdot$ has type $(2)$, i.e., $\cdot : S \times S \rightarrow S$ and $(S', \circ)$, where $\circ$ has type $(3)$, i.e., $\cdot : S' \times S' \times S' \rightarrow S'$ then is it possible that $(S, \cdot)$ and $(S', \circ)$ are isomorphic or $(S', \circ)$ is embed in $(S, \cdot)$?

Answer 1

Your question is ill posed. From a logical point of view it doesn't make any sense to compare two structures which have different signature: as in this case where you have a structure with a binary operation and another one with a ternary operation. That's because there's no easy way to see how to define a morphisms between structures which have different signature.

What could make sense is a comparison of the clones which can be obtained from the structure above. The clones associated to two structure are themselves some sort of structure which can be considered as a sort of closure or completion of the given structures, and for such structures become meaningful make a comparison.

If there's an isomorphism between the clones of two structures then the two structures are two presentation of essentially the same algebraic object: the two presentation are different because one in one case the clone is generated via a binary operation in the other case with a ternary operation. So basically isomorphisms of clones generated by different structures means that the theories (in the logical sense) of the two structures are essentially the same.

An embedding amount to prove that on structure have enough information to codify the other structure inside itself. This means that the theory of one structure is in some sense richer or stronger of the second one.

I hope this help.