Isomorphism

Homomorphisms can be classified according to the type of mapping hbetween the algebras. A homomorphism is an isomorphism if $h : A \rightarrow B$ is bijective. In other words, the elements of A and B can be put into one-to-one correspondence. This means that not only every value in B is mapped to, but there is also an inverse function which will map each value in B to a value in A. If two $\Sigma$-algebras, denoted by the same signature $\Sigma$are isomorphic, they are ``equivalent'' in all respects apart from the name of their sets and operations. Any statement expressed in terms of the function symbols of $\Sigma$ which is true in $\cal{A}$ will also be true of $\cal B$ and vice-versa. Although the algebras may appear quite different in that the elements of the carriers are different, structurally the algebras are identical.

If an isomorphism exists between two algebras, those algebras possess the strongest type of similarity and are equivalent in all respects apart from the the symbols which name the elements of their respective carrier sets. The concept of an isomorphism between algebras is a fundamental one in relation to the specification of abstract data types because it expresses formally the independence of representation inherently necessary for values of abstract data types.