Generalized Residuated And Pseudo R₀- Algebras

Fuzzy logic has been developed quickly, and many new research methods have been appeared with the development by the effort of many scholars for many years. It is one of the important methods in the study of fuzzy logic to introduce t—norms into logic to form the logic systems based on t—norms. The MV—algebras, the Goguen algebras and the Godel algebras are algebraic models for the three significant axiom systems of logic: Lukasiewicz logic, Goguen logic and Godel logic, respectively. These algebras correspond to the three important continuous t—norms on [0,1]: Lukasiewicz t—norms, Goguen t—norms, Godel t—norms, respectively. In [1], Hajek introduced a much abroader axiom systems of BL logic and the BL— algebras corresponding to them, which led that the MV— algebras, the Goguen algebras and the Godel algebras are several important specific examples of BL— algebras. In fact, "⊙" is t—norm on [0,1] and " → " is the associated implication if only " ⊙ " is left continuous. In 1997, Professor Wang Guojun proposed a left continuous t—norm on [0,1], that is, R₀ t—norm [3]. At the same time, Professor Wang introduced R₀— algebra and and axiom systems of L* corresponding to it, and proved the completeness of the axiom by algebraic methods^[2], consequently, these results make the strict foundation for fuzzy reasoning.In 1986, D.Mundici proved the categorical equivalence between MV—algebras and categories of albelian 1—group with strong unit. In 1999, G.Georescu and A.Iorgulescu [9] introduced pseudo MV— algebras as non-commutative extension of MV— algebras. At the same time, A.Dvurecenskij proved that pseudo MV—algebras are equivalent to the categories of 1—group with strong unit. In [5], G.Georgescu and A.Iorgulescu gave the definations and several properties of pseudo t—norms, (weak) pseudo BL algebras and some properties of them. They pointed out that continuous pseudo t—norms on [0,1] are t—norms on [0,1]. Then the left continuous pseudo t—norms on [0,1] become another important object in fuzzy logic. In this paper,a kind of left continuous pseudo t—norms —pseudo Rot—norms and pseudo R₀—algebras based on them are discussed in details.