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|.