Transformations between Algebras

At this point, the bewildering number of different models of a single given signature leads us to wonder

• is there is some kind of ``equivalence'' or ``resemblance'' between the members of the class (family) of algebras which are models of a given signature ?
• is there one member (or group of members) of this class which somehow captures the intrinsic properties of that entire class ?

Before we can address these issues, we need to examine how an algebra can be transformed into another using a mapping between the carrier sets of the algebras. In the following discussion, we will confine ourselves to homogeneous algebras. The ideas developed here extend quite naturally to heterogeneous algebras but the added complexity of notation needed would only serve to obscure the fundamental concepts.

Two algebras ${\cal A} = [A,\Omega_A]$ and ${\cal B} = [B,\Omega_B]$that are denoted by a common signature can certainly said to be similar in the sense that they will have the same number of operations of matching arities which allows the sets of operations $\Omega_A$ and $\Omega_B$ to be put into a one-to-one correspondence. This is a weak form of equivalence and is based purely on a syntactic classification. Nothing is stated about any connection between the semantic properties of the algebras (as prescribed by any equations satisfied by the operations) which defines how the values of the carrier sets A and B are related. We might expect, for example, to be able to map one algebra onto another while preserving the inherent structure of the operations.