In this note, we study of those congruences on an Ockham algebra with de Morgan skeleton that the quotient algebras belong to the class of de Morgan algebras. We particularly give a description of those kernel ideals ...In this note, we study of those congruences on an Ockham algebra with de Morgan skeleton that the quotient algebras belong to the class of de Morgan algebras. We particularly give a description of those kernel ideals that generate these congruences.展开更多
Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a cong...Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ on S,a mapping Γ:σ|→(σ Q,σ K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S,where σ K is the restriction of σ to K and σ Q=(σ∨ρ K)/ρ K. Moreover,the lattice of congruences of S is also discussed. As a special case,every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?展开更多
Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.展开更多
Which subset of an ordered semigroup S can serve as a congruence class of certain order-congruence on S is an important problem. XIE Xiangyun proved that if every ideal C of an ordered semigroup S is a congruence clas...Which subset of an ordered semigroup S can serve as a congruence class of certain order-congruence on S is an important problem. XIE Xiangyun proved that if every ideal C of an ordered semigroup S is a congruence class of one order-congruence on S, then C is convex and when C is strongly convex, the reverse statement is true in 2001. In this paper, we give an alternative constructing order congruence method, and we prove that every ideal B is a congruence class of one order congruence on S if and only if B ...展开更多
In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥...In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥cknlogk for 2 + k), the equation is solved nontrivially in any β-adic field,where β is a prime ideal of K.展开更多
In this paper, the internal structures of MS-algebras are investigated by means of the new notions of Stonean kernel and M-ideal. We determine necessary and sufficient conditions for an MS-algebra with a Kleene skelet...In this paper, the internal structures of MS-algebras are investigated by means of the new notions of Stonean kernel and M-ideal. We determine necessary and sufficient conditions for an MS-algebra with a Kleene skeleton. After introducing a special kind of principal congruence, we consider the relationship between an MS-algebra and its quotient algebra.展开更多
文摘In this note, we study of those congruences on an Ockham algebra with de Morgan skeleton that the quotient algebras belong to the class of de Morgan algebras. We particularly give a description of those kernel ideals that generate these congruences.
文摘Let S be an ideal nil-extension of a completely regular semigroup K by a nil semigroup Q with zero. A concept of admissible congruence pairs (δ,ω) of S is introduced, where δ and ω are a congruence on Q and a congruence on K respectively. It is proved that every congruence on S can be uniquely respresented by an admissible congruence pair (δ,ω) of S. Suppose that ρ K denotes the Rees congruence induced by the ideal K of S. Then it is shown that for any congruence σ on S,a mapping Γ:σ|→(σ Q,σ K) is an order-preserving bijection from the set of all congruences on S onto the set of all admissible congruence pairs of S,where σ K is the restriction of σ to K and σ Q=(σ∨ρ K)/ρ K. Moreover,the lattice of congruences of S is also discussed. As a special case,every congruence on completely Archimedean semigroups S is described by an admissible quarterple of S. The following question is asked: Is the lattice of congruences of the completely Archimedean semigroup a semimodular lattice?
基金supported in part by National Natural Science Foundation of China(Grant No.10701048)Natural Science Foundation of Shandong Province (Grant No.ZR2009AM007)+2 种基金Independent Innovation Foundation of Shandong Universitysupported in part by National Basic Research Program of China (973 Program) (Grant No.2007CB807902)National Natural Science Foundation of China (Grant No.10601034)
文摘Let K be an algebraic number field of finite degree over the rational filed Q.Let ak be the number of integral ideals in K with norm k.In this paper we study the l-th integral power sum of ak,i.e.,∑k≤ x akl(l = 2,3,...).We are able to improve the classical result of Chandrasekharan and Good.As an application we consider the number of solutions of polynomial congruences.
基金the National Natural Science Foundation of China (Nos.10626012 103410020)+2 种基金the Jiangsu Planned for Postdoctoral Research Found (No.0502022B)the Natural Science Foundation of Guangdong Province (No.0501332)the Educational Department Natural Science Foundation of Guangdong Province (No.Z03070)
文摘Which subset of an ordered semigroup S can serve as a congruence class of certain order-congruence on S is an important problem. XIE Xiangyun proved that if every ideal C of an ordered semigroup S is a congruence class of one order-congruence on S, then C is convex and when C is strongly convex, the reverse statement is true in 2001. In this paper, we give an alternative constructing order congruence method, and we prove that every ideal B is a congruence class of one order congruence on S if and only if B ...
文摘In this paper, the additive equations of the type α_1λ_1~k+ … +α_sλ_s^k = 0 are studied, α_i'sbeing integers of an algebraic number field K of degree n. The main result is as follows: Ifs≥(2k)^(n+1) (or s≥cknlogk for 2 + k), the equation is solved nontrivially in any β-adic field,where β is a prime ideal of K.
文摘In this paper, the internal structures of MS-algebras are investigated by means of the new notions of Stonean kernel and M-ideal. We determine necessary and sufficient conditions for an MS-algebra with a Kleene skeleton. After introducing a special kind of principal congruence, we consider the relationship between an MS-algebra and its quotient algebra.
