In connection with the question Is a set together with an operation always a relational structure?,
I am trying to represent an isomorphism of algebraic structures as an isomorphism of relational structures.
Let's say a relational structure is a set with one or more n-ary relation on it.
An n-ary relation on a set $A$ is a non-empty subset of the Cartesian power $A^n$.
Let's say an algebraic structure is a set with one or more n-ary operation on it.
An n-ary operation on a set $A$ is a map of a non-empty subset of the Cartesian power $A^n$ onto $A$.
Clearly, an algebraic structure is a relational structure with the following relations:
- The product relation is the image (a subset of $A^1$) of the map;
- For each element $p$ from the products relation there is also an operand relation which is the preimage of $p$ in the Cartesian power $A^n$ (a set of subsets of $A^n$).
We can define an isomorphism between relational structures as a regular relation-preserving isomorphism.
Then, we can say that two algebraic structures are isomorphic if:
- They are product relation isomorphic, and
- They are operand relation isomorphic for each pair from the product relation isomorphism.
Would it be a correct and an equivalent definition of isomorphism of algebraic structures?