Gr (?) Bner-shirshov Base Theory And Applications

This thesis consists of four parts.In Chapter 1, we recall the Gr(o|¨)bner-Shirshov bases theory and Composition-Diamond Lemma for associative.algebrasoIn Chapter 2, a Gr(o|¨)bner-Shirshov bases for the free product and tensorproduct of associative K-algebras with identities were given, respectively. Bythe Composition-Diamond Lemma, the K-bases of the free product and tensorproduct were obtained. As an application, we prove the Normal Form Theoremfor free product of groups.In Chapter 3, a Gr(o|¨)bner-Shirshov bases for the free product and tensorproduct of associative K-algebras (not necessarily with identity) were given,respectively. The K-bases of the free product and tensor product were obtainedby using the Composition-Diamond Lemma.In Chapter 4, a Gr(o|¨)bner-Shirshov basis of the Chinese monoid CH(X)was obtained. Moreover, by the Composition-Diamond lemma, we have a nor-mal form of words of the Chinese monoid CH(X).