The Study Of Semigroup Algebras

In recent years, the theory of semigroup algebras has been developing very fast, and there are many signi?cant results have been obtained: not only on the classic properties of semigroup algebras, but also on their applications in other?elds. For example, probability, combinatorics, statistical mechanics and topology.There are so many unknown problems on semigroup algebras waiting for us to explore. In the study of the theory of Artian algebras, the determinator of a map plays a very important role.In this paper, we focus on the cellularity of U-semiabundant semigroup algebras, the direct product decomposition and projective indecomposable modules of locally adequate semigroup algebras, the semiprimitivity of orthodox semigroup algebras, the π-semisimplity and primeness of locally inverse semigroup algebras,the classi?cation and representation type of three dimensional semigroup algebras,and the determinator of a map in the module category of a hereditary algebra.In Chapter 2, we provide that the semigroup algebra of the U-semiabundant semigroup with Rees matrix semigroups over monoids as principal ～U-factors is cellular if and only if all of the monoid algebras of its structure monoids are cellular. We also study the cellularity of the semigroup algebra of a semilattice of Rees matrix semigroups. As consequences, we get the cellularity of super abundant semigroup algebras and completely regular semigroup algebras.In Chapter 3, we consider the the direct product decomposition and representation type of locally adequate concordant semigroup algebras. A key step is to construct a multiplicative basis of this semigroup algebra by using Rukola?ne idempotents. This allows us to de?ne a primitive abundant semigroup S, which is with better properties. Then, we can study the original contracted semigroup algebra R0[S] in terms of R0[S]. The main results in this chapter: on the one hand, we decompose this class of locally adequate semigroup algebras into the direct product of certain primitive abundant 0-J-simple semigroup algebras; on the other hand, we provide a description of the projective indecomposable modules in terms of the R-classes of S, and then determine its representation type.Let S be a ?nite orthodox semigroup or an orthodox semigroup with idempotent band locally pseudo?nite. In Chapter 4, we study the semiprimitivity of the contracted semigroup algebra R0[S]. By using the principal factors of S and the Rukola??ne idempotents of R0[S], we prove that R0[S] is semiprimitive if and only if S is an inverse semigroup and for each maximal subgroup G of S, R[G] is semiprimitive. These results strengthen previous results about the semiprimitivity of inverse semigroup algebras.In Chapter 5, we characterize the π-semisimplity and primeness of locally inverse semigroup algebras. Let S be a locally inverse semigroup with idempotent set locally pseudo?nite. By using the semigroup S constructed in Chapter 3, we prove that E(S) is locally ?nite if and only if R0[S] is the direct product of some contracted completely 0-simple semigroup algebras; and in this case, prove that the condition D = J holds in S. Moreover, if we assume that S satis?es the property that D = J, then we provide a characterization of the π-semisimplity of R0[S]. At the end of this section, we study the primeness of R0[S].In Chapter 6, we study the properties of three dimensional semigroup algebras(with or without identity) over an algebraically closed ?eld. We mainly use the concepts of Jacobson radical, the complete set of primitive orthogonal idempotents and quivers of an algebra. Not only all the isomorphism classes of three dimensional semigroup algebras are given, but we also determine their representation type.Note that every representation ?nite algebra with identity can be represented as a contracted semigroup algebra. Then, by using the above results on three dimensional semigroup algebras we have obtained, and the properties on some four dimensional semigroup algebras, all representation ?nite three dimensional algebras(with an identity) can be determined.Let f be a map of a hereditary algebra KQ. In Chapter 7, the correspondence between cokernel functors Ffin the module category KQ and quotients of projective modules of the preprojective algebra ΠQis studied. What we want to show is how to use the socle of some associated quotient ΠQ-module to compute the socle of a cokernel functor in mod KQ. Then we can obtain the determinator of the map f because it is the same as the socle of the coknernel functor Ff.