Algebras of Distributions of Binary Isolating Formulas for Quite $o$-Minimal Theories

Algebras of distributions of binary isolating formulas over a type for quite $o$-minimal theories with nonmaximal number of countable models are described. It is proved that an isomorphism of these algebras for two $1$-types is characterized by the coincidence of convexity ranks and also by simultaneous satisfaction of isolation, quasirationality, or irrationality of those types. It is shown that for quite $o$-minimal theories with nonmaximum many countable models, every algebra of distributions of binary isolating formulas over a pair of nonweakly orthogonal types is a generalized commutative monoid.