For an analogy: an Abelian group is a structure in which four axioms hold. A non-Abelian group is a structure in which three of these axioms hold, and the fourth does not. It is not a structure in which some random proper subset of axioms holds, because such a notion would be useless. While structures where specific subsets of axioms hold (non-Abelian groups, semigroups, monoids, etc.) are useful.