Free Bicommutative Algebras

Abstract

Algebras with identities a(bc) = b(ac) , (ab)c = (ac)b is called bicommutative.Bases and the cocharacter sequence for free bicommutative algebras are found.It is shown that the exponent of the variety of bicommutaive algebras is equal to 2 . Introduction.One of important problems of modern algebra is to study free algebras satisfying some identities.To construct bases as vector spaces, to find the cocharacter sequence, to construct multilinear components and to find their dimensions or asymptotics of the growth are parts of this problem.From this point of view varieties of associative algebras, varieties of associative and commutative algebras and varieties of Lie algebras are well understood.For example, free associative commutative algebras are polynomial algebras.Relations between homogeneous polynomials and symmetric polynomials are used in many