Filters of R0-algebras
نویسندگان
چکیده
In order to research the logical system whose propositional value is given in a lattice from the semantic viewpoint, Xu [7] proposed the concept of lattice implication algebras, and discussed some of their properties. Xu and Qin [8] introduced the notion of implicative filters in a lattice implication algebra, and investigated some of their properties. Turunen [5] introduced the notion of Boolean deductive system, or equivalently, Boolean filter in BL-algebras which rise as Lindenbaum algebras from many valued logic introduced by Hájek [2]. Boolean filters are important because the quotient algebras induced by Boolean filters are Boolean algebras, and a BL-algebra is bipartite if and only if it has proper Boolean filter. In [6], Wang introduced the notion of R0-algebras in order to provide an algebraic proof of the completeness theorem of a formal deductive system. We note that R0-algebras are different from BL-algebras because the identity x∧ y = x (x→ y) holds in BL-algebras, but does not hold in R0-algebras. R0-algebras are also different from lattice implication algebras because the identity (x → y) → y = (y → x)→ x holds in lattice implication algebras, but does not hold in R0-algebras. Although they are different in essence, they have some similarities, that is, they all have the implication operator →. Therefore, it is meaningful to generalize some aspects of lattice implication algebras and BL-algebras to R0-algebras. In [1], Esteva and Godo introduced the MTL-algebra; the MTL-algebra is an extension of a BL-algebra, which is obtained by eliminating the condition x∧ y = x (x → y) in a BL-algebra. In fact, the MTL-algebra is an algebra induced by a left continuous t-norm and its corresponding residuum, but the BL-algebra is an algebra induced by a continuous t-norm and its corresponding residuum. It is proved an that R0-algebra is a particular MTL-algebra and
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عنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2006 شماره
صفحات -
تاریخ انتشار 2006