Monographic Studies Of Ideals/Filters In BL-algebras

Basic Logic which’s propositional truth values in [0,1] is a many-valued logic and an important foundation of fuzzy logic. BL-algebras are the Lindenbaum-Tarski algebras and match basic logic, one motivation for its introduction is to provide an algebraic framework for basic logic, the second one is to offer alge-braic methods for studying continuous t-norm on [0,1]. From an algebraic logic point of view, studies of logic systems either use algebraic methods to research algebraic systems associated with logic systems, or convert logic problems into al-gebraic ones and solve it, then convert the results into logical forms again. In this paper, we study basic logic from an algebraic point of view, it mainly studies some of important issues about the theory of ideals and filters of BL-algebras, including the following topics:(1) Prime ideals and prime ideal space in BL-algebras are investigated.The method of generating an ideal is provided, the representation theorem of ideals is also established and the set (?)(A) of all ideals is proved to be an algebraic lattice, then the necessary and sufficient conditions for (?)(A) is a Boolean alge-bra are given. The prime ideal theorem is established in BL-algebras, we point out that maximal ideals and prime ones exist in any BL-algebras, and maximal ideals are prime, but the converse is not true. Also it turns out that irreducible ideals and prime ideals coincide. In an MV-algebra A, the following is proved to be equiv-alent:(ⅰ){0} ideal is prime, (ⅱ) all proper ideals are prime, (ⅲ) A is linear. The prime representation theorem and minimal associated prime representation theo-rem of ideals are given. Especially, the existence and uniqueness of finite minimal prime representation of ideals in dual NBL-algebras are also proved. Finally, the prime ideal space is a Stone space and the maximal one is a regular space and also a T4-space are proved in BL-algebras.(2) Fuzzy (prime) ideals and falling fuzzy Godel ideals in BL-algebras are investigated.The method of generating a fuzzy ideal by a fuzzy set in BL-algebras is pro-vided, the representation theorem of fuzzy ideals is established, and then the set of all fuzzy ideals is proved to be a complete distributive lattice satisfied infinitely distributive law. It turns out that every fuzzy irreducible ideal must be a fuzzy prime ideal, but the converse is not true and then their equivalent conditions are also provided. The fuzzy prime ideal theorem and the fuzzy prime representation theorem of ideals in BL-algebras are established. Furthermore, falling fuzzy ideals are introduced in BL-algebras and its equivalent characterizations are given. Also Godel ideals are defined, and under certain conditions, it is proved that an BL-algebra is a Godel algebra if and only if {0} ideal is Godel ideal. And then falling fuzzy Godel ideals are introduced in BL-algebras and several equivalent forms are given. Moreover, it is proved that a falling fuzzy Boolean ideal is a falling fuzzy Godel ideal and a falling fuzzy implicative ideal is equivalent to a falling fuzzy Godel ideal. Finally, combining falling implication operators with fuzzy ide-als and fuzzy Godel ideals respectively,I-fuzzy ideals and I-fuzzy Godel ideals are introduced in BL-algebras and their characterizations are also given.(3) The generalized co-annihilators theory of BL-algebras is investigated.The notion of generalized co-annihilators in BL-algebras is introduced and it is proved to be a filter of BL-algebras, then its some important properties are studied. The (relative) involutory filters relative to the filter (F){1} are defined and then the set SF(A) of all involutory filters relative to F is proved to be a complete Boolean lattice, furthermore, the specific form that SF(A) (S(A)) being an BL- algebra is given. Prime filters and minimal (associated) prime filters in BL-algebras are characterized by means of generalized co-annihilators and we point out that P is prime if and only if (P:G)= P (see Theorem 4.34), P is a minimal prime filter associated with F if and only if Fp=P (see Theorem 4.39), P is a minimal prime filter if and only if 1p= P (see Corollary 4.40). A representation theorem of (rela-tive) involutory filters is also provided, the relations between (relative) involutory filters and minimal (associated) prime filters are mainly discussed, then it turns out:an (relative) involutory filter can be represented as the intersection of mini-mal (associated) prime filters and a minimal prime filter then can be represented as the union of involutory filters. Finally, it is proved that a co-annihilator in the quo-tient algebra is the quotient of a generalized co-annihilator, and if an BL-algebra is involutory, then its every quotient algebra is involutory too, etc.