The subset relation and 2-stratified sentences in set theory and class theory

Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of $\mathrm{ZF}$, $\mathrm{BAS}$, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a complete extension, ${\mathrm{IABA}}_{\mathrm{Ideal}}$, of the theory of infinite atomic boolean algebras and a minimum subsystem, ${\mathrm{BAC}}^{+}$, of $\mathrm{NBG}$ are identified with the property that if $M$ is a model of ${\mathrm{BAC}}^{+}$, then $⟨M,S^M,{\subseteq }^M⟩$ is a model of ${\mathrm{IABA}}_{\mathrm{Ideal}}$, where M is the underlying set of $M$, $S^M$ is the unary predicate that distinguishes sets from classes and ${\subseteq }^M$ is the definable subset relation. These results are used to show that that $\mathrm{BAS}$ decides every 2-stratified sentence of set theory and ${\mathrm{BAC}}^{+}$ decides every 2-stratified sentence of class theory.