The Mutual Relations Among BCK-algebras, Residuated Lattices, BR₀-algebras And Some Researches On Opposite Direction Triple I Method

Residuated lattice is a fuzzy logic algebra system which is widely used BCK-algebra and BR₀-algebra are also very important algebra systems. In this paper, by discussing the mutual relations among BCK-algebras, Residuated tattices, BR₀-algebras, we get some significant results; we also get some significant results by discussing both restricted triple I method, opposite direction restricted (?)riple I method under Lukasiewicz implication operator and opposite direction triple I method under complete co-residuated lattice. In the last, the relations graph among BCK-algebras, Residuated lattices, BR₀-algebras are given in the tabulated paper.The construction and the main contents of this paper are as follows:Chapter 1. Preliminary knowledge. In this chapter, we make a depiction for basic knowledge and basic conclusions of BCK-algebras, Residuated lattices, BR₀-algebras which would be used in the following chapters.Chapter 2. By discussing the mutual relations among BCK-algebras, residuated lattices, BR₀-algebras, we get some significant results, such as involution lattice BCK-algebras and regular residuated lattices are equivalent; bounded commutive lattice BCK-algebras and normal residuated lattices are equivalent; MV-algebras and bounded commutive BCK-algebras are equivalent; BR₀-algebras and strong regular residuated lattices are equivalent and so on.Chapter 3. By discussing both restricted triple I method and opposite direction restricted triple I method under Lukasiewicz implication operator, we give out the key to the question of restricted triple I method and opposite restricted triple I method with respect to FMP and FMT models under Lukasiewicz implication operator. By discussing the opposite triple I method under complete co-residuated lattice, we introduce CRL-opposite directed triple I MIFMP formula and CRL-opposite directed triple I MIFMT formula, and we give out the key to question of opposite directed triple I method with respect to FMP and FMT models under complete co-residuated lattice. we also prove opposite directed triple I MIFMP method and opposite directed triple I MIFMT method are reducing methods and so on. In the last, under the operator of Lukasiewicz co-adjoint pairs and R₀ co-adjoint pairs, we give out the key to the question of opposite directed triple I method with respect to MIFMP formula and MIFMT formula.