Some Contributions To Residuated Lattices

The study of the theory of residuated lattices is one of the important topics in algebra.Many experts and scholars over the past decade have carried out thorough,painstaking and systematical research into it.Residuated lattices are indeed partially ordered semigroups.Thus,we can study the residuated lattices by means of the viewpoints and methods in the algebraic theory of semigroups.We mainly study three classes of residuated lattices from the angle of semigroups in this paper.The dissertation consists of four chapters.In Chapter 1,the background, the present state of residuated theory and some fundamental knowledge about semigroups,residuated lattices and universal algebra are simply introduced. In Chapter 2,we study idempotent residuated chains.After obtaining some properties of such residuated lattice,we establish a structure theorem for idempotent residuated chains,which generalizes main results in[1].Moreover, we give necessary and sufficient conditions for a band with an identity to be the semigroup reduct of some idempotent residuated chain.Chapter 3 is devoted to research conical idempotent residuated lattices,which is an extension of the previous chapter and in-depth.Firstly,we obtain some properties of conical lattice-ordered idempotent monoids by using the Green D-relation on the semigroup reducts.Secondly,we establish a structure theorem for conical idempotent residuated lattices.We also give a charaterization of conical idempotent residuated lattices,which generalizes results in[2].Finally,based on the structure theorem,we obtain some characterizations of subdirectly irreducible, simple and strictly simple conical idempotent residuated lattices. In particular,we here give the classification of strictly simple conical idempotent residuated lattices.In Chapter 4,we study E-unitary inverse residuated chains.After giving some properties of such residuated lattices,we obtain a structure theorem for E-unitary inverse residuated chains.As applications, we consider some special cases of E-unitary inverse residuated chains.Finally, we give some characterizations of subdirectly irreducible,simple and strictly simple E-unitary inverse residuated chains.