Consider the two realisations of the signature Numbers, the algebras |Num| and . Even though they are quite different, it is possible to formalise a relationship between them. (The purpose of this becomes clearer later on.)
Consider a function from
the carrier to the carrier
, defined as follows:
so whatever the value of returns a value in the range 0 ...3. We can also define a function between the carriers and :
that is, the identity function. We will refer to the two functions, and together, as f. f is called a homomorphism if it preserves structure. This is best explained by an example.
Consider the operation in the algebra |Num|, with operands 2 and 3 in the carrier . Then
We performed the operation first, then applied the function to the result and got the answer 1. Now if f is a homomorphism, we should be able to apply the function to the operands first, then perform the corresponding operation and obtain the same result, namely, 1. If we try this we get
If this is generally true for all the named operations, then f is a homomorphism and we write . Note that there is not in this case a corresponding or inverse homomorphism from to |Num|.