In this paper, we discuss some propertie s of lattice implication algebra and difine the transitivity of implication in a set, we show the transitivity of implication and the substitution Theorem hold i n filters. S...In this paper, we discuss some propertie s of lattice implication algebra and difine the transitivity of implication in a set, we show the transitivity of implication and the substitution Theorem hold i n filters. So every filter of lattice implication algebra satisfies the Syllogis m and substitution Theorem of propositional logic.展开更多
This paper proved that when we have a binary operation and two nullary operations O and I on a non-empty set L,we can define the operations',V and A on L by these operations such that if these operations satisfy t...This paper proved that when we have a binary operation and two nullary operations O and I on a non-empty set L,we can define the operations',V and A on L by these operations such that if these operations satisfy the seven axioms of lattice implication algebra then(L,V.∧,’)is a com plemented lattice with universal bounds.So,when we define lattice implication algebra,we needn't start on a complemented lattice with universal bounds,but we can begin with a non-empty set with.out any algebraic structure.展开更多
A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-L...A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed.Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated.Moreover,it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.展开更多
In this paper, a topological space based on LI-ideals of a lattice implication algebra is constructed, and its topological properties, such as separability, compactness and connectedness are discussed.
Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper, we introduce the concept of. I-feature filters and involutory filters in fat ti...Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper, we introduce the concept of. I-feature filters and involutory filters in fat tice implication algebras, and discussed some propert ies of them. Finally, the characterization of filters of any lattice implication algebra which satisfies Increasing Chain Conditions (I. C. C) is given.展开更多
Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then...Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras （LIAs）, i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.展开更多
As a continuate work,ideal-based resolution principle for lattice-valued first-order logic system LF(X) is proposed,which is an extension of α-resolution principle in lattice-valued logic system based on lattice impl...As a continuate work,ideal-based resolution principle for lattice-valued first-order logic system LF(X) is proposed,which is an extension of α-resolution principle in lattice-valued logic system based on lattice implication algebra.In this principle,the resolution level is an ideal of lattice implication algebra,instead of an element in truth-value field.Moreover,the soundness theorem is given.In the light of lifting lemma,the completeness theorem is established.This can provide a new tool for automated reasoning.展开更多
In this paper, some necessary and sufficient conditions that a finite lattice implication algebra is simple are established. Specially, it is proved that a finite lattice implication algebra L is simple if and only if...In this paper, some necessary and sufficient conditions that a finite lattice implication algebra is simple are established. Specially, it is proved that a finite lattice implication algebra L is simple if and only if （L, ≤） is a chain, if and only if there exists the unique dual atom in L. Also, it is given that a finite lattice implication algebra with order of a prime number is simple.展开更多
In the present paper, the intuitionistic fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties and equivalent characterizations of intuition- istic fuzzy LI-ideals are given. ...In the present paper, the intuitionistic fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties and equivalent characterizations of intuition- istic fuzzy LI-ideals are given. Representation theorem of intuitionistic fuzzy LI-ideal which iS generated by an intuitionistic fuzzy set is established. It is proved that the set consisting of all intuitionistic fuzzy LI-ideals in a lattice implication algebra, under the inclusion order, forms a complete distributive lattice.展开更多
In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and di...In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and discussed the relation between the category of topological space and the category of lattice implication algebras.展开更多
In the present paper, the interval-valued （ε, εv q）-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued （ε, ε v q）-fuzzy LI-ideals are given. Repres...In the present paper, the interval-valued （ε, εv q）-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued （ε, ε v q）-fuzzy LI-ideals are given. Representation theorem of interval-valued （ε, ε v q）-fuzzy LI-ideal which is generated by an interval-valued fuzzy set is established. It is proved that the set consisting of all interval-valued （ε, εv q）-fuzzy LI-ideals in a lattice implication algebra, under the partial order , forms a complete distributive lattice.展开更多
In order to study uncertainty reasoning and automatic reasoning with linguistic terms, in this paper, the set of basic linguistic truth values and the set of modifiers are defined, according to common sense; partially...In order to study uncertainty reasoning and automatic reasoning with linguistic terms, in this paper, the set of basic linguistic truth values and the set of modifiers are defined, according to common sense; partially orderings are defined on them. Based on it, a lattice implication algebra model L18 of linguistic terms is built; furthermore, its some basic properties are discussed.展开更多
Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discuss...Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discussed,and then proved that both of the categorys of the two algebras are categorical equivalence. Finally,the infinitely distributivity in lattice implication algebras were proved.展开更多
The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-latt...The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-lattice implication algebra as the valuation field,and the syntax and semantic of the logical system are discussed,respectively.展开更多
In this paper,the concepts of product and factorization of lattice implication algebra are proposed,the relation between lattice implication product algebra and its factors and some properties of lattice implication p...In this paper,the concepts of product and factorization of lattice implication algebra are proposed,the relation between lattice implication product algebra and its factors and some properties of lattice implication product algebras are discussed.展开更多
Many-valued logic system always plays a crucial role in artificial intelligence.In orderfurther to study many-valued logic system as well as logic with truth-values in a lattice,the con-cept of lattice implication alg...Many-valued logic system always plays a crucial role in artificial intelligence.In orderfurther to study many-valued logic system as well as logic with truth-values in a lattice,the con-cept of lattice implication algebra was proposed in reference[1]and the corresponding logic systemwas also investigated.In this paper.we focus on a kind of important lattice implication algebra,i.e..injective lattice implication algebra.Some properties are discussed and also the charaeteristic ofits structure is given.展开更多
This paper introduced the concept of L-fuzzy sub lattice implication algebra and discussed its properties. Proved that the intersection set of a family of L-fuzzy sub lattice implication algebras is a L-fuzzy sub latt...This paper introduced the concept of L-fuzzy sub lattice implication algebra and discussed its properties. Proved that the intersection set of a family of L-fuzzy sub lattice implication algebras is a L-fuzzy sub lattice implication algebra, that a L-fuzzy sub set of a lattice implication algebra is a L-fuzzy sub lattice implication algebra if and only if its every cut set is a sub lattice implication algebra, and that the image and original image of a L-fuzzy sub lattice implication algebra under a lattice implication homomorphism are both L-fuzzy sub lattice implication algebras.展开更多
First, we reviewed the definitions of lattice implication algebras, lattice implication subalgebras, and LI-ideals, and provided an equivalent definition of LI-ideal. Then we investigated some properties of lattice im...First, we reviewed the definitions of lattice implication algebras, lattice implication subalgebras, and LI-ideals, and provided an equivalent definition of LI-ideal. Then we investigated some properties of lattice implication subalgebra and U-ideal, and found the least lattice implication subalgebra. Finally, the relation between lattice implication subalgebra and LI-ideal is presented. It is proved that no LI-ideals are non-trivial lattice implication subalgebras.展开更多
In this paper, we defined the concept of implicative and fuzzy implicative ideals of lattice implication algebras, and discussed the properties of them. And then, we pointed out the relations between implicative ideal...In this paper, we defined the concept of implicative and fuzzy implicative ideals of lattice implication algebras, and discussed the properties of them. And then, we pointed out the relations between implicative ideal and LI _ideal, implicative iedal and implicative filter, implicative ideal and fuzzy implicative ideal, fuzzy implicative ideal and fuzzy implicative filter, and fuzzy implicative ideal and fuzzy LI _ideal.展开更多
We have given a semantic extension of lattice-valued propositional logic LP（X） in [6]. In this paper, we investigate its corresponding syntactic extension of LP（X） and give the relations between these two extensions.
文摘In this paper, we discuss some propertie s of lattice implication algebra and difine the transitivity of implication in a set, we show the transitivity of implication and the substitution Theorem hold i n filters. So every filter of lattice implication algebra satisfies the Syllogis m and substitution Theorem of propositional logic.
基金Supported by the National Natural Science Foundation of China(60074014)
文摘This paper proved that when we have a binary operation and two nullary operations O and I on a non-empty set L,we can define the operations',V and A on L by these operations such that if these operations satisfy the seven axioms of lattice implication algebra then(L,V.∧,’)is a com plemented lattice with universal bounds.So,when we define lattice implication algebra,we needn't start on a complemented lattice with universal bounds,but we can begin with a non-empty set with.out any algebraic structure.
基金Supported by the National Natural Science Foundation of China(11501523,61673320)。
文摘A 2-dimension linguistic lattice implication algebra(2DL-LIA)can build a bridge between logical algebra and 2-dimension fuzzy linguistic information.In this paper,the notion of a Boolean element is proposed in a 2DL-LIA and some properties of Boolean elements are discussed.Then derivations on 2DL-LIAs are introduced and the related properties of derivations are investigated.Moreover,it proves that the derivations on 2DL-LIAs can be constructed by Boolean elements.
基金Supported by the National Natural Science Foundation of China(60474022)Supported by the Henan Innovation Project For University Prominent Research Talents(2007KYCX018)
文摘In this paper, a topological space based on LI-ideals of a lattice implication algebra is constructed, and its topological properties, such as separability, compactness and connectedness are discussed.
文摘Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper, we introduce the concept of. I-feature filters and involutory filters in fat tice implication algebras, and discussed some propert ies of them. Finally, the characterization of filters of any lattice implication algebra which satisfies Increasing Chain Conditions (I. C. C) is given.
基金The National Natural Science Founda-tion of China (No.60474022)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.20060613007)
文摘Some new properties of lattice filters are presented based on the order-preserving mapping and lattice homomorphism, and two necessary and sufficient conditions for lattice filters under the chain type are given. Then, the relations between lattice filter and lattice implication algebras （LIAs）, i. e., the relations between lattice filter and LIA-filters, and the related properties are investigated. In addition, three necessary and sufficient conditions for LIA-filters are discussed. The obtained results may serve as some theoretical supports to lattice-valued logical system.
基金the National Natural Science Foundation of China(No.61175055)the Sichuan Key Technology Research and Development Program(No.2011FZ0051)
文摘As a continuate work,ideal-based resolution principle for lattice-valued first-order logic system LF(X) is proposed,which is an extension of α-resolution principle in lattice-valued logic system based on lattice implication algebra.In this principle,the resolution level is an ideal of lattice implication algebra,instead of an element in truth-value field.Moreover,the soundness theorem is given.In the light of lifting lemma,the completeness theorem is established.This can provide a new tool for automated reasoning.
基金Supported by a grant of Natural Science Foundation of Guangdong Province in China（021073）
文摘In this paper, some necessary and sufficient conditions that a finite lattice implication algebra is simple are established. Specially, it is proved that a finite lattice implication algebra L is simple if and only if （L, ≤） is a chain, if and only if there exists the unique dual atom in L. Also, it is given that a finite lattice implication algebra with order of a prime number is simple.
基金Supported by the National Natural Science Foundation of China.(Grant No.60774073)
文摘In the present paper, the intuitionistic fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties and equivalent characterizations of intuition- istic fuzzy LI-ideals are given. Representation theorem of intuitionistic fuzzy LI-ideal which iS generated by an intuitionistic fuzzy set is established. It is proved that the set consisting of all intuitionistic fuzzy LI-ideals in a lattice implication algebra, under the inclusion order, forms a complete distributive lattice.
文摘In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and discussed the relation between the category of topological space and the category of lattice implication algebras.
基金Supported by the National Natural Science Foundation of China(Grant No.60774073)the Higher School Research Foundation of Inner Mongolia(Grant No.NJSY14283)
文摘In the present paper, the interval-valued （ε, εv q）-fuzzy LI-ideal theory in lattice implication algebras is further studied. Some new properties of interval-valued （ε, ε v q）-fuzzy LI-ideals are given. Representation theorem of interval-valued （ε, ε v q）-fuzzy LI-ideal which is generated by an interval-valued fuzzy set is established. It is proved that the set consisting of all interval-valued （ε, εv q）-fuzzy LI-ideals in a lattice implication algebra, under the partial order , forms a complete distributive lattice.
基金Supported by the National Natural Science Foundation of China ( No.60474022)the Specialized Research Fund for the Doctoral Program of Higher Education of China (No.20060613007)
文摘In order to study uncertainty reasoning and automatic reasoning with linguistic terms, in this paper, the set of basic linguistic truth values and the set of modifiers are defined, according to common sense; partially orderings are defined on them. Based on it, a lattice implication algebra model L18 of linguistic terms is built; furthermore, its some basic properties are discussed.
文摘Lattice implication algebras is an algebraic structure which is established by combining lattice and implication algebras. In this paper,the relationship between lattice implication algebras and MV algebra was discussed,and then proved that both of the categorys of the two algebras are categorical equivalence. Finally,the infinitely distributivity in lattice implication algebras were proved.
基金the National Natural Science Foundation of China(No.61175055)the Scientific Research Fund of Sichuan Provincial Education Department(11ZB023)the Sichuan Key Technology Research and Development Program(No.2011FZ0051)
文摘The modal lattice implication algebra(i.e.,M-lattice implication algebra) is introduced and its properties are investigated.The modal lattice-valued propositional logical system is introduced by considering the M-lattice implication algebra as the valuation field,and the syntax and semantic of the logical system are discussed,respectively.
基金Project Supported by the National Natural Science Foundation of China.
文摘In this paper,the concepts of product and factorization of lattice implication algebra are proposed,the relation between lattice implication product algebra and its factors and some properties of lattice implication product algebras are discussed.
基金Supported by the National Science Foundation of China(No.69874033).
文摘Many-valued logic system always plays a crucial role in artificial intelligence.In orderfurther to study many-valued logic system as well as logic with truth-values in a lattice,the con-cept of lattice implication algebra was proposed in reference[1]and the corresponding logic systemwas also investigated.In this paper.we focus on a kind of important lattice implication algebra,i.e..injective lattice implication algebra.Some properties are discussed and also the charaeteristic ofits structure is given.
文摘This paper introduced the concept of L-fuzzy sub lattice implication algebra and discussed its properties. Proved that the intersection set of a family of L-fuzzy sub lattice implication algebras is a L-fuzzy sub lattice implication algebra, that a L-fuzzy sub set of a lattice implication algebra is a L-fuzzy sub lattice implication algebra if and only if its every cut set is a sub lattice implication algebra, and that the image and original image of a L-fuzzy sub lattice implication algebra under a lattice implication homomorphism are both L-fuzzy sub lattice implication algebras.
基金The National Natural Science Foundationof China (No.60875034)the Specialized Research Fundfor the Doctoral Program of Higher Education of China (No.20060613007)
文摘First, we reviewed the definitions of lattice implication algebras, lattice implication subalgebras, and LI-ideals, and provided an equivalent definition of LI-ideal. Then we investigated some properties of lattice implication subalgebra and U-ideal, and found the least lattice implication subalgebra. Finally, the relation between lattice implication subalgebra and LI-ideal is presented. It is proved that no LI-ideals are non-trivial lattice implication subalgebras.
文摘In this paper, we defined the concept of implicative and fuzzy implicative ideals of lattice implication algebras, and discussed the properties of them. And then, we pointed out the relations between implicative ideal and LI _ideal, implicative iedal and implicative filter, implicative ideal and fuzzy implicative ideal, fuzzy implicative ideal and fuzzy implicative filter, and fuzzy implicative ideal and fuzzy LI _ideal.
基金Supported by the National Natural Science Foundation of China(60474022)
文摘We have given a semantic extension of lattice-valued propositional logic LP（X） in [6]. In this paper, we investigate its corresponding syntactic extension of LP（X） and give the relations between these two extensions.
