On algebras of distributions of binary formulas for theories of unars

Algebras of distributions of binary isolating and semi-isolating formulas are derived structures for a given theory. These algebras reflect binary links between realizations of $1$-types defined by formulas of the initial theory. Thus these are two sorts of interrelated classification problems: 1) to define, for a given class of theories, what algebras correspond to theories in this class and to classify these algebras; 2) to classify theories in the class in the dependence of algebras of isolating and semi-isolating algebras that defined by these theories. For the finite algebras of binary isolating formulas that description implies the description for the algebra of binary semi-isolating formulas.
In the paper, we give the description for algebras of distributions of binary isolating formulas for theories of unars with unary predicates, which is based on multiplication tables for these algebras. It is proved that any theory of unar with unary predicates defines, on a set of realizations of $1$-type, an algebra of of distributions of binary isolating formulas, which is obtained by an algebra isomorphic to exactly one of the following algebras: 1) the additive group of integer numbers, 2) a cyclic group, 3) a cyclic algebra with given number of connected components, 4) an algebra of free unar with given number of preimages for each element; 5) the additive monoid of natural numbers; 6) an algebra of low cones. In particular, if the unary function, in the unar, is a substitution, then the algebra of distributions of binary isolating formulas is defined by an algebra isomorphic to exactly one of the following algebras: the additive additive group of integer numbers, a cyclic group, a cyclic algebra with given number of connected components. The structures of these algebras allow to classify initial theories of substitutions. Finite algebras are exhausted by the following list: cyclic groups, cyclic algebras with given number of connected components, algebras of low cones.