Example 3

Suppose that $\Theta$ has the same (principal and auxiliary) base sets as $\Sigma$, and the same typical characterization. If, moreover, the axiom of $\Sigma$ implies (in $\mathscr{T}$) the axiom of $\Theta$, it is clear that the term $s$ is a procedure of deduction of a structure of species $\Theta$ from a structure of species $\Sigma$. $\Theta$ is then said to be poorer than $\Sigma$, and $\Sigma$ is richer than $\Theta$. Every structure of species $\Sigma$, in a theory $\mathscr{T}'$ which is stronger than $\mathscr{T}$, is then also a structure of species $\Theta$. For example, the species of structures of totally ordered sets (obtained by taking as axiom the conjunction of the axiom of order structures (no. 4, Example 1) and the relation $s\cup s^{-1}=A\times A$) is richer than the species of order structures. \*The species of commutative group structures is richer than the species of group structures. The species of compact topological space structures is richer than the species of topological structures, etc. \*