Semigroup method in combinatorics on words

Semigroup method and Combinatorics method are two important techniques in the study of words and languages. This paper presents the semigroup method and explore languages by using semigroup method. Some algebraic properties of languages be given via semigroup method. Firstly, the semigroup constructions of dense languages are given. It's proved that every dense language which syntactic monoid has the zero, or which syntactic monoid has the minimal i-deal and this ideal is a periodic semigroup having a primitive idempotent contains a language w(W^K)* for some nonempty word w and some positive integer k. The theorem every code is thin is generalized to the result that every code which syntactic monoid has the zero, or which syntactic monoid has the minimal ideal and this ideal is a periodic semigroup having a primitive idempotent is thin. It's proved that every dense regular language contains a product of some word with some dense regular right unitary submonoid of A*, and by using this result it is proved that every dense regular language also contains a primitive word. Secondly, the regular component decomposition of free monoids are given. It's proved that free monoids are regular component splittable and that every product of a regular component splittable language with a regular component split-table suffix language is regular component splittable. As applications of the above results two conjectures about regular languages given by Shyr and Yu, in 1998 are proved. The two conjectures are that regular languages are regular component splittable and dense regular languages contains imprimitive words.