I wonder what the minimal axiomatization of a set of structures mean? I came across this term from Wikipedia:
For a theory $T\in A,$ let $F(T)$ be the set of all structures that satisfy the axioms $T$; for a set of mathematical structures $S$, let $G(S)$ be the minimal axiomatization of $S$. We can then say that $F(T) $ is a subset of $S$ if and only if T logically implies $G(S)$: the "semantics functor" $F$ and the "syntax functor" $G$ form a monotone Galois connection, with semantics being the lower adjoint.
My guess is that given a set of structures, its minimal axiomatization is the set of axioms such that the set of all structures satisfying the set of axioms is the given set of structures, isn't it?
Thanks and regards!