ABSTRACT
Monomial algebras form a class of well-behaved graded algebras, with the grading coming from the natural grading in the free semigroup. Finitely generated monomial algebras arise in a natural way in the context of the growth of associative algebras. The study of monomial algebras is motivated by some problems in algebraic topology. Connections of this type and some applications of the combinatorics in free semigroups developed for monomial algebras are given. A special case of so-called path algebras of graphs and homomorphic images of these algebras is of basic importance in the theory of representations of artinian algebras. This chapter characterizes monomial algebras satisfying polynomial identities. It first states two auxiliary results on ideals in the free semigroup. The chapter briefly discusses chain conditions for monomial algebras.
