Isomorphisms

Why

We often have two algebras for which we can identify elements of the ground set.

Let $(A, +_{A})$ and $(B, +_B)$ be two algebras.¹

An isomorphism between these two algebras is a bijection $f: A \to B$ satisfying:

\[ f(a \, +_A \, a') = f(a) \, +_B \, f(a') \]

and

\[ f^{-1}(b +_B b') = f^{-1}(b) +_A f^{-1}(b'). \]

If there exists an isomorphism between two algebras we say that the algebras are isomorphic.

Future editions will change this notation to avoid clashes with right and left identity elements (see Identity Elements). ↩︎