Congruences on Zappa-Szép Products of Semilattices with An Identity and Groups

Let P =E (8) G be a Zappa-Szép product of a semilattice E with an identity and a group G.In this paper,we first introduce the concept of congruence pairs for P,and then prove that every congruence on P...     

Rectangular Ring Congruences on an E-inversive Semiring

In this paper, we discussed the property of rectangular band semiring congruence and ring congruence on a semiring and gave some characterizations and structure of rectangular ring congruence on an E-inversive semiring.     

The Least Regular Order with Respect to a Regular Congruence on Ordered Γ-Semigroups

The motivation mainly comes from the conditions of congruences to be regular that are of importance and interest in ordered semigroups. In 1981, Sen has introduced the concept of the Γ-semigroups. We can see that any semigroup can be considered as a Γ-semigroup. In this paper, we introduce and characterize the concept of the regular congruences on ordered Γ-semigroups and prove the following statements on an ordered Γ-semigroup M : (1) Every ordered semilattice congruences is a regular congruence. (2) There exists the least regular order on the Γ-semigroup M/ρ with respect to a regular congru- ence ρ on M . (3) The regular congruences are not ordered semilattice congruences in general.     

Congruence Pairs of Decomposable MS-Algebras*

In this paper, the authors first introduce the concept of congruence pairs on the class of decomposable MS-algebras generalizing that for principal MS-algebras (see [13]). They show that every congruence relation θ on a decomposable MS-algebra L can be uniquely determined by a congruence pair (θ₁, θ₂), where θ₁ is a congruence on the de Morgan subalgebra L^?? of L and θ₂ is a lattice congruence on the sublattice D(L) of L. They obtain certain congruence pairs of a decomposable MS-algebra L via central elements of L. Moreover, they characterize the permutability of congruences and the strong extensions of decomposable MS-algebras in terms of congruence pairs.     

纯正半群上的同余

Let S be an orthodox semigroup and γ the least inverse congruence on S. C(S) denotes the set of all congruences on S. In this paper we introduce the concept of admissible triples for S, where admissible triples are constructed by the congruences on S/γ. the equivalences on E(S)/L and E(S)/R. The notation Ca(S) denotes the set of all admissible triple for S. We prove that every congruence p on S can be uniquely determined by the admissible triple induced by p, and there exists a lattice isomomorphism between C(S) and Ca(S).     

RESEARCH ANNOUNCEMENTS Block Separating Congruence Lattices . of .E-bisimple Semigroups

In 1966, Reilly[1] Characterized bisimple ω-semigroups as Bruck-Reilly extensions of Groups. Later, Munn and Reilly[2] proved that a congruence on a bisimple ω-semigroup is a group congruence or is idempotent-separating. Many author investigated the congruences on Bruck-Reilly extensions of some semigroups for about twenty years. In this paper, we study the block-separating     

On the Addition of Two Cubes of Units and Nonunits mod p~α

Let p ≡ 2(mod 3) be an odd prime and α be a positive integer. In this paper,for any integer c, we obtain a formula for the number of solutions of the cubic congruence x~3+ y~3≡ c(mod p~α) with x, y units, nonunits and mixed pairs, respectively. We resolve a problem posed by Yang and Tang.     

The K°-relation on the Congruence Lattice of a Regular Semigroup with an Inverse Transversal

Let S be a regular semigroup with an inverse transversal S° and C(S) the congruence lattice of S. A relation K° on C(S) is introduced as follows: if ρ, θ∈ C(S), then we say that ρ and θ are K°-related if Ker ρ° = Ker θ° , where ρ° = ρ|S°. Expressions for the least and the greatest congruences in the same K°-class as ρ are provided. A number of equivalent conditions for K° being a congruence are given.     

On a novel eccentricity-based invariant of a graph

In this paper,for the purpose of measuring the non-self-centrality extent of non-selfcentered graphs,a novel eccentricity-based invariant,named as non-self-centrality number(NSC number for short),of a graph G is defined as follows:N(G)=∑v_i,v_j∈V(G)|e_i-e_j| where the summation goes over all the unordered pairs of vertices in G and e_i is the eccentricity of vertex v_i in G,whereas the invariant will be called third Zagreb eccentricity index if the summation only goes over the adjacent vertex pairs of graph G.In this paper,we determine the lower and upper bounds on N(G) and characterize the corresponding graphs at which the lower and upper bounds are attained.Finally we propose some attractive research topics for this new invariant of graphs.     

The Poincaré Series of Relative Invariants of Finite Pseudo-Reflection Groups

Let F be a field with characteristic 0,V=F~n the n-dimensional vector space over F and let G be a finite pseudo-reflection group which acts on V.Let χ:G→F~* be a 1-dimensional representation of G.In this article we show that X(g)=(detg)~α(0≤α≤r-1),where g∈G and r is the order of g.In addition,we characterize the relation between the relative invariants and the invariants of the group G,and then we use Molien's Theorem of invariants to compute the Poincaré series of relative invariants.     

纯正半群上的同余扩张(一)

刻划半群上的同余及其扩张是半群的代数理论中的一个非常重要的课题．本文讨论了带上的同余的正规性和不变性以及在其Ｈａｌｌ半群上的扩张，从同余扩张的角度刻划了带上的同余的性质，给出了扩张的极大、极小同余的描述．     

含幺逆半群的半格的同余对

本文利用一族含幺逆半群的同余对刻画了其半格的同余对,并给出了含幺逆半群半格的正规同余对族的格与其标准同余对的格之间的同构关系.     

有限群的$p$-覆盖远离子群及$S$-拟正规嵌入子群

Let G be a finite group, p the smallest prime dividing the order of G and P a Sylow p-subgroup of G. If d is the smallest generator number of P, then there exist maximal subgroups P1, P2,..., Pd of P, denoted by Md(P) = {P1,...,Pd}, such that di=1 Pi = Φ(P), the Frattini subgroup of P. In this paper, we will show that if each member of some fixed Md(P) is either p-cover-avoid or S-quasinormally embedded in G, then G is p-nilpotent. As applications, some further results are obtained.     

子群的S-条件置换性对有限群结构的影响

Let G be a finite group. Fix a prime divisor p of IGI and a Sylow p-subgroup P of G, let d be the smallest generator number of P and Ma（P） denote a family of maximal subgroups P1, P2 , Pd of P satisfying ∩^di=1 Pi = Ф（P）, the Frattini subgroup of P. In this paper, we shall investigate the influence of s-conditional permutability of the members of some fixed .Md（P） on the structure of finite groups. Some new results are obtained and some known results are generalized.     

具有弱$s$-置换嵌入子群的有限群

Let P be a Sylow p-subgroup of a group G with the smallest generator number d,where p is a prime.Denote by M_d(P) = {P_1,P_(2,...,)P_d} a set of maximal subgroups of P such that φ(P) = ∩_(n=1)~dP_n.In this paper,we investigate the structure of a finite group G under the assumption that the maximal subgroups in M_d(P) are weakly s-permutably embedded in G,some interesting results are obtained which generalize some recent results.Finally,we give some further results in terms of weakly s-permutably embedded subgroups.     

Congruence triples on regular semigroups and their generalizations

In this paper, we introduce the concept of VT-congruence triples on a regular semigroup S and show how such triples can be constructed by using the equivalences on S/ℒ, S/R and the special congruences on S. Also, such congruence triples are characterized so that an associated congruence can be uniquely determined by a given congruence triple. Moreover, we also consider the VH-congruence pairs on an orthocryptogroup.     

关于有限群的S-半置换子群

Let d be the smallest generator number of a finite p-group P and let Md（P） = {P1,...,Pd} be a set of maximal subgroups of P such that ∩di=1 Pi = Φ（P）. In this paper, we study the structure of a finite group G under the assumption that every member in Md（Gp） is S-semipermutable in G for each prime divisor p of ｜G｜ and a Sylow p-subgroup Gp of G.     

关于广义Stirling数偶的同余性

本文证明了广义Ｓｔｉｒｌｉｎｇ数偶的一些同余性质，从而回答了文［５］中的一个猜测．这些结果做为特例推广了已知的关于两类Ｓｔｉｒｌｉｎｇ数的同余性质．     

High-Dimensional Shape Fitting in Linear Time

Let $P$ be a set of $n$ points in $\Re^d$. The {\em radius} of a $k$-dimensional flat ${\cal F}$ with respect to $P$, which we denote by ${\cal RD}({\cal F},P)$, is defined to be $\max_{p \in P} \mathop{\rm dist}({\cal F},p)$, where $\mathop{\rm dist}({\cal F},p)$ denotes the Euclidean distance between $p$ and its projection onto ${\cal F}$. The $k$-flat radius of $P$, which we denote by ${R^{\rm opt}_k}(P)$, is the minimum, over all $k$-dimensional flats ${\cal F}$, of ${\cal RD}({\cal F},P)$. We consider the problem of computing ${R^{\rm opt}_k}(P)$ for a given set of points $P$. We are interested in the high-dimensional case where $d$ is a part of the input and not a constant. This problem is NP-hard even for $k = 1$. We present an algorithm that, given $P$ and a parameter $0 < \eps \leq 1$, returns a $k$-flat ${\cal F}$ such that ${\cal RD}({\cal F},P) \leq (1 + \eps) {R^{\rm opt}_k}(P)$. The algorithm runs in $O(nd C_{\eps,k})$ time, where $C_{\eps,k}$ is a constant that depends only on $\eps$ and $k$. Thus the algorithm runs in time linear in the size of the point set and is a substantial improvement over previous known algorithms, whose running time is of the order of $d n^{O(k/\eps^c)}$, where $c$ is an appropriate constant.     

Modularity prevents tails

We establish a direct correspondence between two congruence properties for finite algebras. The first property is that minimal sets of type have empty tails. The second property is that congruence lattices omit pentagons of type .