In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we o...In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.展开更多
In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations an...In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].展开更多
Let S be an antinegative commutative semiring without zero divisors and Mn(S) be the semiring of all n × n matrices over S. For a linear operator L on Mn(S), we say that L strongly preserves nilpotent matrice...Let S be an antinegative commutative semiring without zero divisors and Mn(S) be the semiring of all n × n matrices over S. For a linear operator L on Mn(S), we say that L strongly preserves nilpotent matrices in Mn(S) if for any A ∈ Mn(S), A is nilpotent if and only if L(A) is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.展开更多
Here we introduce a subclass of the class of Ockham algebras (L; f) for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn(x) and f^n+1(x) are complementary. We characterize ...Here we introduce a subclass of the class of Ockham algebras (L; f) for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn(x) and f^n+1(x) are complementary. We characterize the structure of the lattice of congruences on such an algebra (L; f). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.展开更多
It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand...It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand, a recently proposed measure-based fuzzy logic(MBFL) satisfies all the axioms of Boolean algebra. In this paper, a complete and thorough proof is given for this.展开更多
Measure based fuzzy logic, which is constructed on the basis of eight axioms, is a seemingly powerful fuzzy logic. It possesses several remarkable properties. (1) It is an extended Boolean logic, satisfying all the p...Measure based fuzzy logic, which is constructed on the basis of eight axioms, is a seemingly powerful fuzzy logic. It possesses several remarkable properties. (1) It is an extended Boolean logic, satisfying all the properties of Boolean algebra, including the law of excluded middle and the law of contradiction. (2) It is conditional. Conditional membership functions play an important role in this logic. (3) The negation operator is not independently defined with the conjunction and disjunction operators, but on the contrary, it is derived from them. (4) Zadehs fuzzy logic is included in it as a particular case. (5) It gives more hints to the relationship between fuzzy logic and probability logic.展开更多
A formal technique for incorporating two specification paradigms is presented,in which an algebraic specifi- cation is implemented by a set of abstract procedures specified in pre- and post-condition style.The link be...A formal technique for incorporating two specification paradigms is presented,in which an algebraic specifi- cation is implemented by a set of abstract procedures specified in pre- and post-condition style.The link be- tween the two level specifications is provided via a translation from terms of algebraic specifications into tempo- ral logic formulae representing abstract programs.In terms of translation,a criterion for an abstract implementation satisfying its specification is given,which allows one to check the consistency between the two levels of specifications.The abstract implementations can be refined into executable code by refining each abstract procedure in it.It is proved that the satisfication relation between a specification and its implementations is preserved by such refinement steps.展开更多
We have studied a function-lock strategy for all-optical logic gate (AOLG) utilizing the cross-polarization modulation (CPM) effect in a semiconductor optical amplifier (SOA). By monitoring the power of logic li...We have studied a function-lock strategy for all-optical logic gate (AOLG) utilizing the cross-polarization modulation (CPM) effect in a semiconductor optical amplifier (SOA). By monitoring the power of logic light, the strategy realized controllable methods to capture OR and NOR functions and switch between them. The strategy has been successfully applied in experiment with 10-Gb/s not-return-to-zero (NRZ) signals, which has a high success-rate above 95% and ensures the high extinction ratio of result light above 11.4 dB. Every step in the strategy has definite numeric evaluation, which provides the potential of automatic implementation.展开更多
To simplify the process for identifying 12 types of symmetric variables in Boolean functions, we propose a new symmetry detection algorithm based on minterm expansion or the truth table. First, the order eigenvalue ma...To simplify the process for identifying 12 types of symmetric variables in Boolean functions, we propose a new symmetry detection algorithm based on minterm expansion or the truth table. First, the order eigenvalue matrix based on a truth table is defined according to the symmetry definition of a logic variable. By analyzing the constraint conditions of the order eigenvalue matrix for 12 types of symmetric variables, an algorithm is proposed for identifying symmetric variables of the Boolean function. This algorithm can be applied to identify the symmetric variables of Boolean functions with or without don't-care terms. The proposed method avoids the restriction by the number of logic variables of the graphical method, spectral coefficient methods, and AND-XOR expansion coefficient methods, and solves the problem of completeness in the fast computation method. The algorithm has been implemented in C language and tested on MCNC91 benchmarks. The application results show that, compared with the traditional methods, the new algorithm is an optimal detection method in terms of the applicability of the number of logic variables, the Boolean function including don't-care terms, detection type, and complexity of the identification process.展开更多
文摘In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.
基金Supported by a grant of National Natural Science Foundation of China(12001243,61976244,12171294,11961016)the Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-762,2021JQ-580)。
文摘In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].
文摘Let S be an antinegative commutative semiring without zero divisors and Mn(S) be the semiring of all n × n matrices over S. For a linear operator L on Mn(S), we say that L strongly preserves nilpotent matrices in Mn(S) if for any A ∈ Mn(S), A is nilpotent if and only if L(A) is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.
文摘Here we introduce a subclass of the class of Ockham algebras (L; f) for which L satisfies the property that for every x ∈ L, there exists n 〉 0 such that fn(x) and f^n+1(x) are complementary. We characterize the structure of the lattice of congruences on such an algebra (L; f). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.
文摘It is well known that Zadeh's fuzzy logic is not a Boolean algebra because it does not satisfy the Law of Excluded Middle or the Law of Contradiction, but the two-valued propositional logic does. On the other hand, a recently proposed measure-based fuzzy logic(MBFL) satisfies all the axioms of Boolean algebra. In this paper, a complete and thorough proof is given for this.
文摘Measure based fuzzy logic, which is constructed on the basis of eight axioms, is a seemingly powerful fuzzy logic. It possesses several remarkable properties. (1) It is an extended Boolean logic, satisfying all the properties of Boolean algebra, including the law of excluded middle and the law of contradiction. (2) It is conditional. Conditional membership functions play an important role in this logic. (3) The negation operator is not independently defined with the conjunction and disjunction operators, but on the contrary, it is derived from them. (4) Zadehs fuzzy logic is included in it as a particular case. (5) It gives more hints to the relationship between fuzzy logic and probability logic.
基金This project is supported by National Natural Science Foundation of China.
文摘A formal technique for incorporating two specification paradigms is presented,in which an algebraic specifi- cation is implemented by a set of abstract procedures specified in pre- and post-condition style.The link be- tween the two level specifications is provided via a translation from terms of algebraic specifications into tempo- ral logic formulae representing abstract programs.In terms of translation,a criterion for an abstract implementation satisfying its specification is given,which allows one to check the consistency between the two levels of specifications.The abstract implementations can be refined into executable code by refining each abstract procedure in it.It is proved that the satisfication relation between a specification and its implementations is preserved by such refinement steps.
基金This work was supported by the National Natural Science Foundation of China under Grant No.60520130298.
文摘We have studied a function-lock strategy for all-optical logic gate (AOLG) utilizing the cross-polarization modulation (CPM) effect in a semiconductor optical amplifier (SOA). By monitoring the power of logic light, the strategy realized controllable methods to capture OR and NOR functions and switch between them. The strategy has been successfully applied in experiment with 10-Gb/s not-return-to-zero (NRZ) signals, which has a high success-rate above 95% and ensures the high extinction ratio of result light above 11.4 dB. Every step in the strategy has definite numeric evaluation, which provides the potential of automatic implementation.
基金supported by the National Natural Science Foundation of China(Nos.61471314 and 61271124)the Zhejiang Provincial Natural Science Foundation(No.LY13F010001)the National Key Technology R&D Program of China(Nos.2013BAH27F01,2013BAH27F02,and 2013BAH27F03)
文摘To simplify the process for identifying 12 types of symmetric variables in Boolean functions, we propose a new symmetry detection algorithm based on minterm expansion or the truth table. First, the order eigenvalue matrix based on a truth table is defined according to the symmetry definition of a logic variable. By analyzing the constraint conditions of the order eigenvalue matrix for 12 types of symmetric variables, an algorithm is proposed for identifying symmetric variables of the Boolean function. This algorithm can be applied to identify the symmetric variables of Boolean functions with or without don't-care terms. The proposed method avoids the restriction by the number of logic variables of the graphical method, spectral coefficient methods, and AND-XOR expansion coefficient methods, and solves the problem of completeness in the fast computation method. The algorithm has been implemented in C language and tested on MCNC91 benchmarks. The application results show that, compared with the traditional methods, the new algorithm is an optimal detection method in terms of the applicability of the number of logic variables, the Boolean function including don't-care terms, detection type, and complexity of the identification process.