Generating Functions for Ternary Algebras and Ternary Trees

摘要

The main object of study are ternary algebras, i.e., algebras with a trilinear operation. In this class we study finitely generated algebras and their growth, as well as the growth of codimensions of absolutely free algebras and some other varieties. For these purposes we use ordinary generating functions and exponential generating functions (the complexity functions). In the classes of absolutely free, free symmetric, free antisymmetric, and some other algebras we study left nilpotent and completely left nilpotent algebras and varieties. The obtained results are equivalent to the enumeration of ternary trees which contain no forbidden subtrees of a special kind. As the main result, we prove that the complexity functions of the varieties of completely left nilpotent and left nilpotent ternary algebras are algebraic.