WBR ₀ - Algebra Equivalent Characterizations And Weakened

Studying algebraic system is an important research direction of many-valued logic, different many-valued logic systems correspond to different many-valued log-ic algebras. After famous logician C.C.Chang proposed MV-algebras, Professor Wu Wangming defined FI-algebras, Professor Xu Yang proposed Lattice implication algebras, Professor P.Hajek established BL-algebras, Professor Wang Guojun estab-lished R0-algebras which are different from BL-algebras. Through the research about R0-algebras by Professor Wu Hongbo, BR0-algebras and WBR0-algebras have been introduced. In this paper, the WBR0-algebras have been studied in depth.the construction and the detailed contents of this paper are as follows:Chapter1:Preliminaries. In this chapter,we give the basic concepts of FI-algebras, WBR0-algebras, posets, Pt-norms (t-norms on posets) and Ps-norms (s-norms on posets) which will be used in this paper.Chapter2:Representation of WBR0-algebras by Pt-norms and Ps-norms. Firstly, the form of Pt-norms (t-norms on posets) on WBRo-algebras is given; Sec-ondly, the equivalence representation of WBR0-algebras has been obtained on posets though Pt-norms and implication operators; Finally, it is proved that the binary op-eration (?) in the original definition of the WBR0-algebras is the Ps-norms(s-norm on posets) of WBR0-algebras, and the equivalence representation of WBR0-algebras has been obtained on posets though Ps-norms and implication operators.Chapter3:The2-kinds of weak forms of WBR0-algebras and their proper-ties. First of all, by weakening the conditions of the WBRo-algebras with2-kinds of different forms, the fuzzy BR0-algebras (FBR0-algebras) and the regular BRo-algebras (RBR0-algebras) are built and then their properties are discussed. Then, it is proved that RBR0-algebras are FBR0-algebras, FBR0-algebras have the same algebraic structure with FI-algebras and RBR0-algebras have the same algebraic structure with regular Fl-algebras. Finaly, it is also showed that FBR0-algebras are not RBRo-algebras and RBRo-algebras are not WBR0-algebras by constructing a example respectively, which explains that there exist the implicational relation and relative independence between3-kinds of algebras.