Br < Sub > 0 < / Sub > Single Atom In Algebra And Corresponding Logic System To Generate True Value Function Characteristics Of The Formula

Non-classical logic is a theoretical basis of fuzzy reasoning and fuzzy control. With non-classical logic gradually coming to maturation and perfection, many scholars have posed various logical algebras based on different implication operators, such as MV-algebras, Fl-algebras, and BRo-algebras. In this paper, based on BRo-algebras, we give some necessary and sufficient conditions for BRo-algebras to be Boolean algebras. Making use of algebraic tools, we explore the relationships betweeen BRo-algebras and other various algebras. We also discuss characterizations of truth value functions of{(?),â†'}-type formulas generated by single atom in BL*system which corresponds to BRo-algebras. By these researches which instill energy into the development of fuzzy logic, especially the theory of fuzzy logic algebra, we could recognize and grasp the common essential features of many logic algebra systems.Chapter I, as preliminaries, introduces basic concepts of partial order, lattices, Heyting-algebras, Fl-algebras and other useful results related to them.Chapter II introduces implicative BRo-algebras, positive implicative BRo-algebras and Heyting-type BRo-algebras. It is proved that the three kinds of BRo-algebras are equivalent to each other and they are all commutative. We also give some necessary and sufficient conditions for BRo-algebras to be Boolean algebras. As a consequent, Boolean algebras are just Heyting-type BRo-algebras. In addition, we represent supremum and infimum of elements’in commutative BRo-algebras by implication operators.In Chapter III, we discuss the relationships between commutative BRo-algebras with regular HFI-algebras, BCK-algebras and semisimple Nelson-algebras. It is proved that commutative BRo-algebras and bounded commutative BCK-algebras are equivalent. Besides, we also give two new characterizations of commutative BRo-algebras, simplifying the conditions for commutative BRo-algebras.In Chapter IV, we discussed characterization of truth value functions induced by{(?),â†'}-type formulas generated by single atom in BL*system. Six special formulas of{(?),â†'}-type and their truth value functions generated by single atom are given. It is proved that the number of truth value functions of single atom formulas is48and that the functions can simply obtained by the six special functions and the identify function via operations of (?) andâ†'.