The Study Of Weak Dually Residuated Lattice Ordered Semigroups

The outstanding feature of the research of fuzzy logics is mutual penetration and merger of logics and algebras, and the powerful algebraic methods have become the main vehicle of the research of fuzzy logics. In a contrary manner, the developments of fuzzy logics have opened the brand-new research areas for the algebras.Previous reseach showed that algebras of fuzzy logics can be viewed as particular cases of bounded commutative DRL-semigroups. The WDRL-semigroups were proposed on the foundation of DRL-semigroups. In this paper, the properties of WDRL-semigroups and the relation between WDRL-semigroups and MTL-algebras were discussed.The ideals of bounded WDRL-semigroups were also studied; simultaneously, the local bounded WDRL-semigroups and the homomorphism of the bounded WDRL-semigroups were researched.The structure and the primary contents of this article are as follows:In chapter one, the research background, the prime task of this paper and the preparation knowledges were mainly introduced.In chapter two, the properties of WDRL-semigroups on the foundation of the existing properties were discussed. Then the unary operation- and the binary operations (?),→in WDRL-semigroups were defined, under which the properties of WDRL-semigroups were also discussed. Simultaneously, the binary operations +,- in MTL-algebras were also defined, under which the properties of MTL-algebras were studied. Further more; the relation between WDRL-semigroups and MTL-algebras was obtained: WDRL-semigroups were categorically equivalent to MTL-algebras under some conditions.In chapter three, the ideals of the bounded WDRL-semigroups and the equal form were defined; several kinds of ideals of bounded WDRL-semigroups as the prime ideals, maximal ideals and generated ideals were discussed. What's more, the relations and natures of them were studied; the conceptions of the local bounded WDRL-semigroups were raised, and the necessary and sufficient conditions when bounded WDRL-semigroups become local bounded WDRL-semigroups were discussed, and so on. At last, the concepts of the congruence, homomorphism and annihilator of WDRL-semigroups were introduced, and some conclusions with ideals related were obtained.