Example 1

Take $\mathscr{T}$ to be the theory of sets, and consider the species of structures which has no auxiliary base set, one principal base set $\mathrm{A}$, the typical characterization $s \in \mathfrak{P}(\mathrm{A} \times \mathrm{A})$, and the axiom

($\Delta_{\mathbf{A}}$ being the diagonal of $\mathrm{A} \times \mathrm{A}$), which is a transportable relation with respect to the typification $s \in \mathfrak{P}(\mathrm{A} \times \mathrm{A})$, as is easily verified. It is clear that the theory of this species of structures is just the theory of ordered sets (Chapter III, §1, no. 3); and therefore the species of structure so defined is also called the species of order structures on $\mathrm{A}$. In Chapter III we saw many examples of sets endowed with structures of this species.