ABSTRACT
This chapter discusses a wide class of semigroup-graded semigroups and algebras to give some general techniques coming from gradations, which are especially useful for the theory of semigroup algebras. It provides some important examples of gradations to establish a strong connection between the theory of graded semigroups and graded algebras. The chapter presents a useful general result on semigroup algebras of semilattices of semigroups. It proves that for a graded monoid, the resulting subalgebra is a contracted semigroup algebra. The chapter discusses the main application of the smash product techniques and also gives a result on the noetherian property of graded algebra.
