E (s) - Right Cancellative Monoid Graph Expansion

Victoria Gould gives us the graph expansion of a right cancellative monoid and attains the construction of a left adequate semigroup from studying a Cayley graph of a right cancellative monoid using *-Green relations. Gracinada M.S.Gomes and Victoria Gould study the graph expansion of an unipotent monoid and show us the construction of a weakly left ample semigroup from studying a Cayley graph of an unipotent monoid using～-Green relations.Many specialists had attained a lot of results in studying right cancellative monoids, unipotent monoids . For example, Let (X, f, S) be a monoid presentation. Then M = M(X, f, S) is left abundant if and only if S is right cancellative; and so on.It is a well-known fact that the Green's -relations differs from Green's *-relations and Green's～-relations. Studying the construction of an E(S)-right cancellative monoid using Green's—relations plays an important role in the research of quasiadequate semigroups.First, this paper give the defination of the E(S)-right cancellative monoid S. Second, we define a congruenceσand we can show that S/σis an E(S/σ)-right cancellative monoid. Third, we obtain the construction of the secondary left ample semigroups using the Cayley graph of a monoid presentation of E(S)-right cancellative monoid and give initial and terminal object of category of secondary left ample monoids. At finally, we give two functions F~σand F~e and show F~σis a adjoint of F~e.There are four charpers of this paper, the main content are as follows: The first charpter is the introduction. It is mainly about the background of this paper. The second charpter introduces the elementary definitions and lemmas which are used in our paper. The third charpter is composed of the principal conclusions of our paper and it is devided into three sections. In the first section we investigate the graph expansions. The second section discusses the category PSLA(X, f, S). The third section gives the functors F~σ and F~e.