On Bijections and Isomorphisms

We list some facts concerning bijections and isomorphisms between algebras for my learning (“algebra” in the universal algebra sense here, not the specific vector space based algebraic structure.)

Theorem 1. If $A$ is an algebra and $A$ is in bijection with $B$ , we may use this bijection to define corresponding operations on $B$ , where $B$ becomes an algebra and the bijection is an isomorphism.

Thus, if a particular bijection is taken to be canonical, we may also immediately consider that to be the canonical isomorphism. (In our usage, the word “canonical” means that we declare a particular bijection to be canonical, and we can use this bijection to replace elements of one set by the other. Any other bijections by contrast are not considered canonical.)

Generally, when we talk about isomorphisms between algebras (as a question different from the bijections between them), we are talking about the isomorphisms between them when the operations have already been defined on both.

Theorem 2. If $i$ is an isomorphism of algebras from $A$ to $B$ , then the set of all isomorphisms from $A$ to $B$ is equal to the set $\left\{ i \circ a \middle| a \in A_{A} \right\}$ , which is also equal to the set $\left\{ b \circ i \middle| b \in A_{B} \right\}$ , where $A_{S}$ is the automorphism group of the algebra $S$ .

This theorem allows identifying one isomorphism between $A$ and $B$ as canonical and reducing the problem of finding the other isomorphisms to classifying the automorphisms of either $A$ or $B$ (which are themselves in bijection, as we would expect since $A$ and $B$ are isomorphic.) This can serve as some motivation for why automorphism groups can be so helpful (as seen in Galois theory.)

Written September 2020.

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