Suppose we have an algebraic specification in the form: $\{S,F,w\}$ where $S$ are the sorts, $F$ are the functions and $w$ are the set of equations. For example, the specification for natural numbers:
- $S = \{\mathrm{int}\}$
- $F = \{ 0: \mathrm{int}, \; \mathrm{succ} : \mathrm{int}\rightarrow\mathrm{int}, \; \mathrm{pred}: \mathrm{int}\rightarrow\mathrm{int} \}$
- $w = \{ \mathrm{succ}(\mathrm{pred}(x)) = x, \; \mathrm{pred}(\mathrm{succ}(x)) = x \}$
My question is, why and where do we need homomorphisms and isomorphisms in this case? How do homomorphisms and isomorphisms look like between algebras ?