A Study On The Ideal And Properties Of BR ₀ - Algebras

In1958, professor Chang.C.C proved the completeness of Lukasiewicz propo-sition logical system by introducing MV algebra, so many logical scholars paid widespread attention to the researches on some logical algebraic structures. In the present, logical algebra as valued range corresponding to the proposition logical system has been an important component part in the content of logical researches. In1997, fuzzy logical system L*and R0-algebra which is related to system L*in semantic respect were established by professor Wang Guo-jun. In1988, through re-searching on Lukasiewicz proposition logic, Godel proposition logic, product propo-sition logic, the basic logical system BL and BL-algebra adjusted to it were proposed by professor Petr Hajek. In2000, according to the studies on R0-algebra and L*system, basic fuzzy proposition logical system BL*and BRo-algebra matched with it were introduced by professor Wu Hong-bo. Both the well known MV algebra and the Ro-algebra are BR0-algebra which is different from BL algebra, so researches on BR0-algebra have an important and theoretical signification, possess universal use value. In this paper, the*ideal is defined by introducing*operation in BR0-algebra. Moreover, the*ideal and its properties are discussed.The construction of chapters and the concrete contents of this paper are as follows:Chapter1:Preliminaries. We give the basic concepts and related theorems of lattice, topological space, BR0-algebra, BR0-subalgebra, which will be used in this paper.Chapter2:The*ideal and its extension theorem of BR0-algebra. At first, the*ideal, prime*ideal, generate*ideal and maximal*ideal are all defined by introducing*operation. Some properties related to those ideals are discussed respectively. Then*ideal extension theorem of BR0-algebra are given. Namely, in a non-degenerate BR0-algebra, any proper*ideal can be extended to a maximal, prime*ideal.Chapter3:The topological space induced by*ideal in BR0-algebra. Firstly, A topological space with the set of all*ideals as its basis on a BR0-algebra is in-troduced. Secondly, characterization of differentiate, closure and interior operations are obtained. Finally, the connection and covering-compaction and separation of this topological space are investigated.Chapter4:The quotient algebra induced by*ideal and BRo-homomorphism. A congruence relation is structured based on the notion of*ideal. It is proved that the quotient algebra of a BR0-algebra under this congruence relation is still a BR0-algebra. What’s more, the definition of BRo-homomorphism is given, and the properties related to it are discussed. At the last, the fundamental theorems of homomorphisms of BR0-algebra are obtained.