The Relations Between Residuated Lattice And Several Algebraic Systems Based On Residuated Lattice

This paper discusses the relations between residuated lattices and several algebraic systems, such as MV-algebras, lattice implication algebras, fuzzy implication algebras, Heyting algebras, Boolean algebras, implication lattices and R0-algebras, ect. as well as the mutual relations among these algebraic systems. Fuzzy MP filters of R0-algebra and RL-type implication operators and triple-I algorithm in fuzzy reasoning are also investigated, of which much attention is paid by many scholars in the field of non-classical logic and fuzzy reasoning. This paper can be divide into four chapters.In non-classical logic, residuated lattice introduced by J.Pavelka is not only an important and fundamental algebraic structures, but also a popular theory and method nowadays. But it seems that this method has not been thoroughly employed in related studies at present. Furthermore, definitions of different algebraic systems, such as MV-algebras, presented by C.C.Chang, fuzzy implication algebras by Wu Wangming, lattice implication algebras by Xu Yang and R0-algebras by Wang Guo-jun etc. have different forms. For this reason, the relations between each one of these algebraic systems and residuated lattice, as well as their important mutual connections are vague, and this brings much inconvenience to the scholars. Therefore, it is important for grasping the mutual connections of these algebraic systems to clarify the relations between each one of them and residuated lattice, which will promote the research in future undoubtedly. In the first chapter of this article, the concepts and the fundamental properties of residuated lattice are introduced. The following three groups of important additional conditions on residuated lattice:are discussed. Some important residuated lattices can be obtained if these conditions have been added to residuated lattice. For example, normal residuated lattice is a kind of residuated lattice satisfied all the above additional conditions; BL-algebra is a kind of residuated lattice satisfied all the conditions in (B) and (C); sub-BL-algebra is kind of residuated lattice satisfied the conditions in (C); a kind of residuated lattice is called sub-normal residuated lattice if it satisfies the conditions in (B). In this paper, the mutual relations of these additional conditions are investigated systematically and clarified. It is proved that each one of these conditions implies distributivity. Furthermore, the mutual relations of normal residuated lattices, subnormal residuated lattices, BL-algebras and sub-BL-algebras and that they are all distributive lattice are known.In the second chapter of this paper, the relations of MV-algebras, lattice implication algebras, Heyting algebras, fuzzy implication algebras, implication lattices, Boolean algebras, (weak) R0-algebras and residuated lattices are discussed, respectively. It is proved that MV-algebras, lattice implication algebras and normal fuzzy implication algebras all are algebraic systems equivalent to normal residuated lattices. It is also proved that weak R0-algebras and regular sub-BL-algebras are equivalent. It is also proved that Boolean algebra is a normal residuated lattice, and that a Heyting algebras is a sub-normal residuated lattices. The necessary and sufficient conditions under which a implication lattice or a regular fuzzy implication algebra is a (regular) residuated lattice are obtained. In the last section of this chapter, the mutual relations of the above various algebras are discussed.Professor Wang Guojun presented and studied the formal system L* of fuzzy logic prepositional calculus. Taking the L*-Lindebaum algebras as background, he established R0- algebras corresponding to L* system. There appear some results on the research of the filters; ideals and congruence relations of R0-algebras. In the third chapter of this article, the concepts of fuzzy MP filters and fuzzy prime MP filters of R0-algebras are introduced. Some characterizations of fuzzy MP filters and fuzzy prime MP filters of R0 algebras are obtained. From t...