| The different fuzzy logic systems are corresponding to the different logic algebras. As far back as 1958, the famous logician C.C.Chang had introduced the theory of MV-algebra and succeeded in proving the completeness of Lukasiewicz fuzzy logic system for solving the completeness of Lukasiewicz fuzzy logic system.In nearly half a century, the scholars of various countries have already made the rich achievement to the studying of MV- algebra and a lot of algebra systems with logic background. These research results have not only promoted the development of fuzzy logic, but also made the content of algebra abundant.The theory of residuated lattices is an important tool to study logic algebras. For example MV- algebra is a special kind of residuated lattice with abundant structures. The primitive definition of MV-algebra had a group amounts to six operators ((?),(?),→, (?), ∨, ∧). Many documents about MV-algebra still refered to operator (?):a(?)b = (a→ b)(?). In those documents, the couple operators (?), (?) which were regarded as association operators were included in the frame of residuated lattice theory. This paper proposes that ((?), (?))as independent operators forms one new structure and is called co-adjoint pairs, which satisfies(T0) (?) : P × P → P is increasing.(S0) (?) : P × P → P is increasing about the first variable and decreasing about the second variable.(C) c ≤ a (?) b iff c (?) b ≤ a, a, b, c ∈ P.Now, the struture((?), (?)) can be introduced in simgroup lattices,and the theory of co-residuated lattices is proposed. In fact, (?) is t-conorm if L=[0,l]. Therefore, (L, (?), (?)) antithesis is in (L, (?), →) and co-adjoint pairs((?), (?)) can be regarded as the natural popularization of (+,-)on lattice. The theory of co-residuated lattices which is put forward in the paper has offered another strong tool for studying fuzzy logic, and has settled new foundation for studying the connection between various kinds of logic algebra.The content of this text is divided into four chapters altogether: Chapter one has provided the preliminary knowledge of the lattice theory that will be used... |
