SWBR ₀ - Algebra And Weak Completeness

In logic algebras, the implication operator is crucial and fundamental in the structure of the logic algebras, and WBRo-algebras is basic algebras that is defined by the implication operator and the (?) operator, and the non-order characteristic of WBRo-algebras provides an approach for setting structures of relative logic algebras and studying relationship between logic algebras. By means of the non-ordered char-acteristic of WBRo-algebras, the structure of SWBRo-algebras is set up by reduction of regularness of WBRo-algebras. Then the concepts of the implication ideal and the prime implication ideal in the structure are introduced by the the implication operator. Moreover, the properties and the applications of the implication ideal are discussed.The construction of chapters and the concrete contents of this paper are as follows:Chapter1:Preliminaries.we give the basic concepts of lattice, residuate lattice, WBR0-algebra, SWBRo-algebra which will be used in this paper.Chapter2:The properties of SWBRo-algebra. At first, the basic properties of SWBRo-algebra are discussed. Then, it is proved that SWBRo-algebras is WBR0-algebras. And the improved form of WBRo-algebra is obtained.Chapter3:The impliction ideal of SWBRo-algebra and its generating method. Firstly, the concept of the implication ideal of SWBR0-algebras is introduced. Then, the generating method of the implication ideal is given in the SWBR0-algebra.Chapter4:The relationship between the implication ideal and the congruence relation. Firstly, the concept of the congruence relation is defined. Secondly, the implication ideal derived by the congruence relation, the congruence relation derived by the implication ideal are discussed. Thirdly, the reducibility and invariability of the implication ideal and the congruence are derived by each other.Chapter5:The weak complete property of SWBR0-algebra. The quotient alge-bra of SWBRo-algebras is defined with the help of the implication idea, furthermore fundamental theorems of homomorphisms of SWBRo-algebra are obtained. Then, the prime implication ideal is defined in the SWBR0-algebras, and the theorem of conditional embedding of SWBR0-algebra is gained by use of the prime implication ideal. Moreover, the weak complete theory of SWBR0-algebra is proved.