Algebras of binary formulas for compositions of theories

We consider algebras of binary formulas for compositions of theories both in the general case and as applied to $\aleph_0$-categorical, strongly minimal, and stable theories, linear preorders, cyclic preorders, and series of finite structures. It is shown that $e$-definable compositions preserve isomorphisms and elementary equivalence and have basicity formed by basic formulas of the initial theories. We find criteria for $e$-definable compositions to preserve $\aleph_0$-categoricity, strong minimality, and stability. It is stated that $e$-definable compositions of theories specify compositions of algebras of binary formulas. A description of forms of these algebras is given relative to compositions with linear orders, cyclic orders, and series of finite structures.