The study of binary algebraic structures (or binary structures) and isomorphisms is a basic (and fundamental) one in the study of abstract algebra, and in some sense, the connection between these two concepts is similar to (in fact, extends) the one between sets and bijection. This post explores that connection.

Recall, two sets A and B have the same “size”, or have an equal number of elements, if there exists a bijection (or bijective function) f: A \to B, i.e. f is a mapping that is both one-to-one and onto. If A and B are both finite sets, then it is easy to prove if there exists a bijection between the two. However, if A and B are infinite sets, then it is usually a non-trivial task proving the existence of a bijection (if there exists one!) between the two sets.

One way of looking at this is by viewing the “structure” of sets as an “obstruction.” So, for instance, we could ask, what could obstruct two sets A and B from having a bijection f: A\to B? Answer: if they have different cardinalities (which may be viewed as a structural property). Okay, I admit the answer is somewhat circular, but we will stick with this for now. Now, is cardinality the only obstruction to the existence of a bijection between any two sets? It turns out the answer is yes. In other words, two sets have the same cardinality if and only if there exists a bijection between the two. Now, let’s extend this further to binary algebraic structures and isomorphisms.

Let’s quickly go through some definitions, first.

1) A binary operation \, * on a set S is a function mapping S \mbox{x} S into S.

2) A binary algebraic structure <S, *> is a set S together with a binary operation \, * on S.

3) Let <S, *> and <S', *'> be binary algebraic structures. An isomorphism of S with S' is a one-to-one and onto function \phi: S \to S' such that

\phi (x * y) = \phi (x)*' \phi (y) \quad \forall x, y \in S.

We note above that the notion of isomorphism between binary algebraic structures “extends” the notion of bijection between sets, in the sense that isomorphism tells us how similar two binary structures are. Just as the existence of a bijection between sets A and B tells us they have the same cardinality (which may be considered a structural property), the existence of an isomorphism between binary structures also tells us how similar they are “structurally.”

Now, just as we asked, earlier, what possible obstructions there might be to the existence of a bijection between two sets, we may ask in a similar vein, what possible obstructions might there be to the existence of an isomorphism between two binary structures <S, *> and <S, *'>? Answer: there is more than one. Let us take a look at some of those obstructions.

1) Cardinality of the sets S and S'.

If S and S' have different cardinalities, then it is easy to prove that there does not exist an isomorphism \phi: S \to S', i.e. <S, *> and <S', *'> are not isomorphic. For example, <\mathbb{Q}, +> and < \mathbb{R}, +> (where + is the usual addition) are not isomorphic because \mathbb{Q} has cardinality \aleph_{0} while |\mathbb{R}| \ne \aleph_{0}. Note that it is not enough to say that \mathbb{Q} is subset of \mathbb{R}. For example, <2\mathbb{Z}, +> is isomorphic to <\mathbb{Z}, +> (where + is the usual addition) even though 2\mathbb{Z} is a subset of \mathbb{Z}. (Here, 2\mathbb{Z} = \{ 2n \mid n \in \mathbb{Z} \}.)

(More to come …)

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