Example 1

\* Let $\Sigma$ be the species of commutative group structures; $\Sigma$ has a single (principal) base set $A$, and its generic structure consists of a single letter $F$; the typical characterization of $\Sigma$ is $F\in\mathfrak{P}((A\times A)\times A)$, and we denote the axiom of $\Sigma$ by $R\{A,F\}$. This axiom implies in particular that $F$ is the graph of a function (the “law of composition” of the group; cf. no. 4, Example 2). In the theory $\mathscr{T}_{\Sigma}$ (where $\mathscr{T}$ denotes the theory of sets) we define a term $M\{A,F\}$ which is a functional graph in $\mathfrak{P}((\mathbf{Z}\times A)\times A)$ and satisfies the following relation $B\{M,A,F\}$:

and

and

and

(“multiplication of an element of $A$ by an integer”).

Consider the species $\Theta$ of $\mathbf{Z}$-module structures, which has a single principal base set $A$, with $\mathbf{Z}$ as auxiliary set, and whose generic structure contains two letters G, L, with the typical characterization

$G \in \mathfrak{P}((A \times A) \times A)$ and $L \in \mathfrak{P}((\mathbf{Z} \times A) \times A)$

and the axiom

“$R \mid A, G$ and $(L$ is a functional graph) and $B \mid L, A, G$”.

It is immediately verified that the terms F, M constitute a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$, and that the term G is a procedure of deduction of a structure of species $\Sigma$ from a structure of species $\Theta$. Furthermore, the condition (3) above is trivially satisfied. We may therefore say that the species of commutative group structures and the species of $\mathbf{Z}$-module structures are equivalent.