In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and con...In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.展开更多
A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorph...A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorphism and δ an α- derivation of a ring R. We prove that （1） If R is an （α, δ）-compatible and weakly 2-primal ring, then R[x; α, δ] is weakly semicommutative; （2） If R is （α, δ）-compatible, then R is weakly 2-primal if and only if R[x; α, δ] is weakly 2-primal.展开更多
Let R be an（α,δ）-compatible ring.It is proved that R is a 2-primal ring if and only if for every minimal prime ideal P in R[x;α,δ] there exists a minimal prime ideal P in R such that P = P [x;α,δ],and that f（...Let R be an（α,δ）-compatible ring.It is proved that R is a 2-primal ring if and only if for every minimal prime ideal P in R[x;α,δ] there exists a minimal prime ideal P in R such that P = P [x;α,δ],and that f（x） ∈ R[x;α,δ] is a unit if and only if its constant term is a unit and other coefficients are nilpotent.展开更多
In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (199...In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999).Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results.In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan's theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive.Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter's theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.展开更多
On the basis of Reiter’s default theory and Zhang Mingyi’s auto compatible default theory, a research on the characters of clausal default theory, especially the closed auto compatible default theory, is carried out...On the basis of Reiter’s default theory and Zhang Mingyi’s auto compatible default theory, a research on the characters of clausal default theory, especially the closed auto compatible default theory, is carried out. First, the theorem of monotonicity with extension number is presented. Second, the proof theory of normal default theory on auto compatible default theory is extended. Some important results are proposed.展开更多
In this paper,we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras.Explicitly,we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Car...In this paper,we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras.Explicitly,we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras.Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras.In particular,we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang(2001)in their study of bi-Hamiltonian structures.Finally,we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.展开更多
In this paper, some important results on the existence and uniqueness of extensions of general default theories are given, and the notion of compatible subset of defaults and that of auto-compatible default are introd...In this paper, some important results on the existence and uniqueness of extensions of general default theories are given, and the notion of compatible subset of defaults and that of auto-compatible default are introduced. Based on the theory developed here, characterizations of extensions of Reiter’s default logic arid Brewka’s cumulative default logic have been obtained respectively, and a class of defaults, the so-called autocompatible defaults, has been presented.展开更多
Receatly, Giordano and Martelli ( 1994) proposed two new cumu- lative variants of Reiter's default logic (DL) : Commitment to Assumptions Default Logic (CADL) and Quasi-Default Logic (QDL). They have only given qu...Receatly, Giordano and Martelli ( 1994) proposed two new cumu- lative variants of Reiter's default logic (DL) : Commitment to Assumptions Default Logic (CADL) and Quasi-Default Logic (QDL). They have only given quasi-inductive characterizations of extensions for the two variants. In this paper, finite characteri- zatioas of extensions for CADL and QDL by applying notions of (joint) compatibility are presented respectively. And corresponding algorithms and complexity results for reasoning are obtairfed.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11071062)the Scientific Research Fundation of Hunan Provincial Education Department(Grant No.12B101)
文摘In this paper we introduce a concept,called Σ-associated primes,that is a generalization of both associated primes and nilpotent associated primes.We first observe the basic properties of Σ-associated primes and construct typical examples.We next describe all Σ-associated primes of the Ore extension R[x; α,δ],the skew Laurent polynomial ring R[x,x-1; α] and the skew power series ring R[[x; α]],in terms of the Σ-associated primes of R in a very straightforward way.Consequently several known results relating to associated primes and nilpotent associated primes are extended to a more general setting.
基金The NSF(11071097,11101217)of Chinathe NSF(BK20141476)of Jiangsu Province
文摘A weakly 2-primal ring is a common generalization of a semicommutative ring, a 2-primal ring and a locally 2-primal ring. In this paper, we investigate Ore extensions over weakly 2-primal rings. Let α be an endomorphism and δ an α- derivation of a ring R. We prove that （1） If R is an （α, δ）-compatible and weakly 2-primal ring, then R[x; α, δ] is weakly semicommutative; （2） If R is （α, δ）-compatible, then R is weakly 2-primal if and only if R[x; α, δ] is weakly 2-primal.
基金Supported by the Natural Foundation of Shandong Province in China(Grant No.ZR2013AL013)
文摘Let R be an（α,δ）-compatible ring.It is proved that R is a 2-primal ring if and only if for every minimal prime ideal P in R[x;α,δ] there exists a minimal prime ideal P in R such that P = P [x;α,δ],and that f（x） ∈ R[x;α,δ] is a unit if and only if its constant term is a unit and other coefficients are nilpotent.
文摘In 1930 Szpilrajn proved that any strict partial order can be embedded in a strict linear order. This theorem was later refined by Dushnik and Miller (1941), Hansson (1968), Suzumura (1976), Donaldson and Weymark (1998), Bossert (1999).Particularly Suzumura introduced the important concept of compatible extension of a (crisp) relation. These extension theorems have an important role in welfare economics. In particular Szpilrajn theorem is the main tool for proving a known theorem of Richter that establishes the equivalence between rational and congruous consumers. In 1999 Duggan proved a general extension theorem that contains all these results.In this paper we introduce the notion of compatible extension of a fuzzy relation and we prove an extension theorem for fuzzy relations. Our result generalizes to fuzzy set theory the main part of Duggan's theorem. As applications we obtain fuzzy versions of the theorems of Szpilrajn, Hansson and Suzumura. We also prove that an asymmetric and transitive fuzzy relation has a compatible extension that is total, asymmetric and transitive.Our results can be useful in the theory of fuzzy consumers. We can prove that any rational fuzzy consumer is congruous, extending to a fuzzy context a part of Richter's theorem. To prove that a congruous fuzzy consumer is rational remains an open problem. A proof of this result can somehow use a fuzzy version of Szpilrajn theorem.
文摘On the basis of Reiter’s default theory and Zhang Mingyi’s auto compatible default theory, a research on the characters of clausal default theory, especially the closed auto compatible default theory, is carried out. First, the theorem of monotonicity with extension number is presented. Second, the proof theory of normal default theory on auto compatible default theory is extended. Some important results are proposed.
基金supported by National Natural Science Foundation of China (Grant Nos.11901501,11922110 and 11931009)supported by the Fundamental Research Funds for the Central Universities+2 种基金Nankai Zhide Foundationsupported by the National Key Research and Development Program of China (Grant No.2021YFA1002000)the Fundamental Research Funds for the Central Universities (Grant No.2412022QD033)。
文摘In this paper,we give Maurer-Cartan characterizations as well as a cohomology theory for compatible Lie algebras.Explicitly,we first introduce the notion of a bidifferential graded Lie algebra and thus give Maurer-Cartan characterizations of compatible Lie algebras.Then we introduce a cohomology theory of compatible Lie algebras and use it to classify infinitesimal deformations and abelian extensions of compatible Lie algebras.In particular,we introduce the reduced cohomology of a compatible Lie algebra and establish the relation between the reduced cohomology of a compatible Lie algebra and the cohomology of the corresponding compatible linear Poisson structures introduced by Dubrovin and Zhang(2001)in their study of bi-Hamiltonian structures.Finally,we use the Maurer-Cartan approach to classify nonabelian extensions of compatible Lie algebras.
基金Project supported by the National Advanced Research and Development
文摘In this paper, some important results on the existence and uniqueness of extensions of general default theories are given, and the notion of compatible subset of defaults and that of auto-compatible default are introduced. Based on the theory developed here, characterizations of extensions of Reiter’s default logic arid Brewka’s cumulative default logic have been obtained respectively, and a class of defaults, the so-called autocompatible defaults, has been presented.
文摘Receatly, Giordano and Martelli ( 1994) proposed two new cumu- lative variants of Reiter's default logic (DL) : Commitment to Assumptions Default Logic (CADL) and Quasi-Default Logic (QDL). They have only given quasi-inductive characterizations of extensions for the two variants. In this paper, finite characteri- zatioas of extensions for CADL and QDL by applying notions of (joint) compatibility are presented respectively. And corresponding algorithms and complexity results for reasoning are obtairfed.
