For Boolean-valued algebraic systems of set-theoretic signature, the notions of transitivity, regularity, and σ-regularity are studied. The notion of a universe over an arbitrary extensional Boolean-valued system is introduced. A description is proposed of the structure of the universe by means of various hierarchies. The results are used for proving the uniqueness of a Boolean-valued universe up to a unique isomorphism.

Keywords:Boolean-valued algebraic system, set theory, Boolean-valued analysis, universe, cumulative hierarchy.