含幺元的半格和群的Zappa-Sz\'{e}p积上的同余

Let P = E G be a Zappa-Szp 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 can be described by such a congruence pair. In fact the congruence lattice on P is lattice-isomorphic to the set of all congruence pairs for P . Finally,we characterize group congruences 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.     

关于具有$Q$-逆断面的正则半群的同余格

In this paper, a complete congruence on the congruence lattice of regular semigroups with Q-inverse transversals is analysed. The classes of this complete congruence which are intervals are discussed and their least and greatest elements are presented clearly.     

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.     

Congruence subgroups of Hecke groups

Hecke groups are an important tool in subgroups of Hecke groups play an important rule investigating functional equations, and congruence in research of the solutions of the Dirichlet series. When q, m are two primes, congruence subgroups and the principal congruence subgroups of level m of the Hecke group H（√q） have been investigated in many papers. In this paper, we generalize these results to the case where q is a positive integer with q ≥ 5, √q ￠ Z and m is a power of an odd prime.     

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).     

A Biordered Set Representation of Regular Semigroups

In this paper, for an arbitrary regular biordered set E, by using biorder-isomorphisms between the w-ideals of E, we construct a fundamental regular semigroup WE called NH-semigroup of E, whose idempotent biordered set is isomorphic to E. We prove further that WE can be used to give a new representation of general regular semigroups in the sense that, for any regular semigroup S with the idempotent biordered set isomorphic to E, there exists a homomorphism from S to WE whose kernel is the greatest idempotent-separating congruence on S and the image is a full symmetric subsemigroup of WE. Moreover, when E is a biordered set of a semilattice Eo, WE is isomorphic to the Munn-semigroup TEo; and when E is the biordered set of a band B, WE is isomorphic to the Hall-semigroup WB.     

Block Separating Congruence Lattices of E－bisimple Semigroups

In 1966, Reilly Characterized bisimple ω-semigroups as Bruck-Reilly extensions of Groups.Later, Munn and Reilly 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 ex-     

圈和完全图的点可区别VE-全染色(英文)

Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.     

σ-多项式的根

§ 1　IntroductionAll graphs considered are finite and simple.Undefined notation and terminology willconform to those in[1 ] .Let V(G) ,E(G) and Gdenote the vertex set,edge set and complement of a graph G,respectively. Let P(G,x) andσ(G,x) denote the chromatic polynomial andσ-polynomialof G,respectively.The log-concavity property of the chromatic polynomial andσ-polyno-mial of G has a close relation to their roots,which were well studied in[2 ,3] .Results onthe study of the roots of…     

A lower bound on the total signed domination numbers of graphs

Let G be a finite connected simple graph with a vertex set V(G)and an edge set E(G). A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1}.The weight of f is W(f)=∑_(x∈V)(G)∪E(G))f(X).For an element x∈V(G)∪E(G),we define f[x]=∑_(y∈NT[x])f(y).A total signed domination function of G is a function f:V(G)∪E(G)→{-1,1} such that f[x]≥1 for all x∈V(G)∪E(G).The total signed domination numberγ_s~*(G)of G is the minimum weight of a total signed domination function on G. In this paper,we obtain some lower bounds for the total signed domination number of a graph G and compute the exact values ofγ_s~*(G)when G is C_n and P_n.     

Trees with Given Diameter Minimizing the Augmented Zagreb Index and Maximizing the ABC Index

Let G be a simple connected graph with vertex set V(G) and edge set E(G).The augmented Zagreb index of a graph G is defined asAZI(G) =∑uv∈E(G)(d_ud_v/(d_u + d_v-2))~3,and the atom-bond connectivity index(ABC index for short) of a graph G is defined asABC(G) =∑uv∈E(G)((d_u + d_v-2)/d_ud_v),where d_u and d_v denote the degree of vertices u and v in G,respectively.In this paper,trees with given diameter minimizing the augmented Zagreb index and maximizing the ABC index are determined,respectively.     

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     

关于广义三项指数和的四次幂均值

The main purpose of this paper is using estimates for trigonometric sums and properties of congruence to study the computation of one kind of fourth power mean of a generalized three-term exponential sum, and give an interesting identity for it.     

On the Factorizations of （mg, mf）-graph

Let G be an (mg, mf)-graph, where g and f are integer-valued functions defined on V(G) and such that 0≤g(x)≤f(x) for each x ∈ V(G). It is proved that(1) If Z ≠ , both g and f may be not even, G has a (g, f)-factorization, where Z = {x ∈ V(G): mf(x)-dG(x)≤t(x) or dG(x)-mg(x)≤ t(x), t(x)= f(x)-g(x)>0}.(2) Let G be an m-regular graph with 2n vertices, m≥n. If (P1, P2,..., Pr) is a partition of m, P1 ≡ m (mod 2), Pi ≡ 0 (mod 2), i = 2,..., r, then the edge set E(G) of G can be parted into r parts E1 , E2,...,Er of E(G) such that G[Ei] is a Pi-factor of G.     

Enumerating reflexible 2-cell embeddings of connected graphs

Two 2-cell embeddings:X → S and j:X → S of a connected graph X into a closed orientable surface S are congruent if there are an orientation-preserving surface homeomorphism h on S and a graph automorphism γ of X such that h = γj.A 2-cell embedding:X → S of a graph X into a closed orientable surface S is described combinatorially by a pair(X;ρ) called a map,where ρ is a product of disjoint cycle permutations each of which is the permutation of the darts of X initiated at the same vertex following the orientation of S.The mirror image of a map(X;ρ) is the map(X;ρ 1),and one of the corresponding embeddings is called the mirror image of the other.A 2-cell embedding of X is reflexible if it is congruent to its mirror image.Mull et al.[Proc Amer Math Soc,1988,103:321-330] developed an approach for enumerating the congruence classes of 2-cell embeddings of graphs into closed orientable surfaces.In this paper we introduce a method for enumerating the congruence classes of reflexible 2-cell embeddings of graphs into closed orientable surfaces,and apply it to the complete graphs,the bouquets of circles,the dipoles and the wheel graphs to count their congruence classes of reflexible or nonreflexible(called chiral) embeddings.     

EQUIVARIANT SELF EQUIVALENCES OF PRINCIPAL FIBRE BUNDLES

Let E be a compact Lie group, G a closed subgroup of E, and H a closed normal sub-group of G. For principal fibre bundle (E,p, E,/G;G) tmd (E/H,p‘,E/G;G/H), the relation between auta(E) (resp. autce (E)) and autG/H(E/H) (resp. autGe/H(E/H)) is investigated by using bundle map theory and transformation group theory. It will enable us to compute the group JG(E) (resp. SG(E)) while the group J G/u(E/H) is known.     

On p-nilpotency and minimal subgroups of finite groups

We call a subgroup H of a finite group G c-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ core(H). In this paper it is proved that a finite group G is p-nilpotent if G is S4-free and every minimal subgroup of P n GN is c-supplemented in NG(P), and when p = 2 P is quaternion-free, where p is the smallest prime number dividing the order of G, P a Sylow p-subgroup of G. As some applications of this result, some known results are generalized.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

On E-solid Regular Semigroups With Inverse Transversals

Let S be a regular semigroup. An inverse subsemigroup S of S is called an inverse transversal if So contains an unique inverse x" for each x E S. In this case, I ~ {e E E(S)lee = e}and A = {g 6 E(S)gg = g}, are left and right regular subbands of S, respectively, where E(S)denotes the set of idempotehts in S. Denote E' = E(S). A regular semigroup S is calledE-solid if R|E(S) o LIE(S) = LIE(S) o R|E(s). A regular semigroup S is E-solid if and only ifthere is an inverse congruence…