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1.
A quasigroup satisfying the 2-variable identity x(yx) = y is called semisymmetric. It is observed that the transpose of a semisymmetric quasigroup is also semisymmetric. Consequently, the existence of a self-orthogonal semisymmetric quasigroup (SOSQ) gives rise to a pair of orthogonal semisymmetric quasigroups. In this paper, the spectrum of SOSQs is investigated and it is found that the spectrum contains all positive integers n = 1 (mod 3), except n = 10.  相似文献   

2.
Let F(x,y) be the free groupoid on two generators x and y. Define an infinite class of words in F(x,y) by w0(x,y) = x,w1(x,y) = y and wi+2(x,y) = wi(x,y)wi+1(x,y). An identity of the form w3n(x,y) = x is called a cyclic identity and a quasigroup satisfying a cyclic identity is called a cyclic quasigroup. The most extensively studied cyclic quasigroups have been models of the identity y(xy) = x. The more general notion of cyclic quasigroups was introduced by N.S. Mendelsohn. In this paper a new construction for cyclic quasigroups is given. This construction is useful in constructing large numbers of nonisomorphic quasigroups satisfying a given cyclic identity or a consequence of a cyclic identity. The construction is based on a generalization of A. Sade's singular direct product of quasigroups.  相似文献   

3.
4.
Let X=(x1, ..., xn) and Y=(y1, ..., ym) be independent samples from populations Gx and Gy, x(1) ,... x(n) be ordered statistics constructed from the sample X. A model of trials associated with the occurrence of dependent events Ak={yk (x(i)}, x(j), i < j, k=1, 2, ..., m, where x(i), x(j) are order statistics, is considered. This model is a generalization of the Bernoulli model. Distribution of frequencies of occurrences of events Ak and the limit theorems which describe asymptotic properties of these frequencies are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 518–528, April, 1990.  相似文献   

5.
Let R be a ring, A = M n (R) and θ: AA a surjective additive map preserving zero Jordan products, i.e. if x,yA are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: AA is a Jordan homomorphism. The third author is Corresponding author.  相似文献   

6.
An example is given of a ringR (with 1) satisfying the standard identityS 6[x 1, ...,x 6] butM 2(R), the 2 × 2 matrix ring overR, does not satisfyS 12[x 1, ...,x 12]. This is in contrast to the caseR=M n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS 2n [x 1, ...,x 2n] andM 2(R) satisfiesS 4n [x 1, ...,x n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.  相似文献   

7.
Letn>0 be an element of the setN of nonnegative integers, and lets(x)=x 1+...+x n , forx=(x 1, ...,x n ) N n . Adiagonal polynomial order inN n is a bijective polynomialp:N n N (with real coefficients) such that, for allx,y N n ,p(x)<p(y) whenevers(x)<s(y). Two diagonal polynomial orders areequivalent if a relabeling of variables makes them identical. For eachn, Skolem (1937) found a diagonal polynomial order. Later, Morales and Lew (1992) generalized this polynomial order, obtaining a family of 2 n–2 (n>1) inequivalent diagonal polynomial orders. Here we present, for eachn>0, a family of (n – 1)! diagonal polynomial orders, up to equivalence, which contains the Morales and Lew diagonal orders.  相似文献   

8.
De Bruijn and Erdős proved that ifA 1, ...,A k are distinct subsets of a set of cardinalityn, and |A i A j |≦1 for 1≦i<jk, andk>n, then some two ofA 1, ...,A k have empty intersection. We prove a strengthening, that at leastk /n ofA 1, ...,A k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440  相似文献   

9.
A class function φ on a finite group G is said to be an order separator if, for every x and y in G \ {1}, φ(x) = φ(y) is equivalent to x and y being of the same order. Similarly, φ is said to be a class-size separator if, for every x and y in G\ {1}, φ(x) = φ(y) is equivalent to |C G (x)| = |C G (y)|. In this paper, finite groups whose nonlinear irreducible complex characters are all order separators (respectively, class-size separators) are classified. In fact, a more general setting is studied, from which these classifications follow. This analysis has some connections with the study of finite groups such that every two elements lying in distinct conjugacy classes have distinct orders, or, respectively, in which disctinct conjugacy classes have distinct sizes. Received: 10 April 2007  相似文献   

10.
Konrad Engel 《Combinatorica》1984,4(2-3):133-140
LetP be that partially ordered set whose elements are vectors x=(x 1, ...,x n ) withx i ε {0, ...,k} (i=1, ...,n) and in which the order is given byxy iffx i =y i orx i =0 for alli. LetN i (P)={x εP : |{j:x j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsxy, x ≯ y, and there exists az εN 1(P) such thatzx andzy. Let ? be the set of all vectorsf=(f 0, ...,f n ) for which there is an Erdös-Ko-Rado familyF inP such that |N i (P) ∩F|=f i (i=0, ...,n) and let 〈?〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and \(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈?〉.  相似文献   

11.
Very Asymmetric Marking Games   总被引:1,自引:0,他引:1  
We investigate a competitive version of the coloring number of a graph G = (V, E). For a fixed linear ordering L of V let s (L) be one more than the maximum outdegree of G when G is oriented so that xy if x < L y. The coloring number of G is the minimum of s (L) over all such orderings. The (a, b)-marking game is played on a graph G = (V, E) as follows. At the start all vertices are unmarked. The players, Alice and Bob, take turns playing. A play consists of Alice marking a unmarked vertices or Bob marking b unmarked vertices. The game ends when there are no remaining unmarked vertices. Together the players create a linear ordering L of V defined by x < y if x is marked before y. The score of the game is s (L). The (a, b)-game coloring number of G is the minimum score that Alice can obtain regardless of Bob’s strategy. The usual (1, 1)-marking game is well studied and there are many interesting results. Our main result is that if G has an orientation with maximum outdegree k then the (k, 1)-game coloring number of G is at most 2k + 2. This extends a fundamental result on the (1, 1)-game coloring number of trees. We also construct examples to show that this bound is tight for many classes of graphs. Finally we prove bounds on the (a, 1)-game coloring number when a < k.  相似文献   

12.
The following theorem is proved. Let n be a positive integer and q a power of a prime p. There exists a number m = m(n, q) depending only on n and q such that if G is any residually finite group satisfying the identity ([x 1,n y 1] ⋯ [x m,n y m ])q ≡ 1, then the verbal subgroup of G corresponding to the nth Engel word is locally finite.  相似文献   

13.
Let R be a ring, A = M n (R) and θ: AA a surjective additive map preserving zero Jordan products, i.e. if x,yA are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains \frac12\frac{1}{2} and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: AA is a Jordan homomorphism.  相似文献   

14.
For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.  相似文献   

15.
Imagine that randomly oriented objects in the shape of a regularn-sided polygon are moving on a conveyor. Our aim is to specify sequences composed of two different rigid motions which, when performed on these objects, will reposition them in all possible ways. We call such sequencesfacing sequences. (Expressed in group theoretical terms, a facing sequence in a groupG is a sequence of elementsa 1,a 2, ...,a n fromG such thatG={e,a 1,a 1 a 2, ...,a 1 a 2 ...a n }). In this paper we classify various kinds of facing sequences and determine some of their properties. The arguments are group theoretical and combinatorial in nature.  相似文献   

16.
A Stein groupoid (quasigroup) is a groupoid (quasigroup) satisfying the identityx(xy)=yx. We show that, for certain two variable identities, the variety of Stein groupoids defined by any one of these identities has the properties that every groupoid in the variety is a quasigroup and that the free groupoid generated by two elements is of finite (small) order which we determine. These results provide characterizations of some Stein quasigroups of small order and we give some further characterizations involving other identities.  相似文献   

17.
Diperfect graphs     
Gallai and Milgram have shown that the vertices of a directed graph, with stability number α(G), can be covered by exactly α(G) disjoint paths. However, the various proofs of this result do not imply the existence of a maximum stable setS and of a partition of the vertex-set into paths μ1, μ2, ..., μk such tht |μiS|=1 for alli. Later, Gallai proved that in a directed graph, the maximum number of vertices in a path is at least equal to the chromatic number; here again, we do not know if there exists an optimal coloring (S 1,S 2, ...,S k) and a path μ such that |μ ∩S i|=1 for alli. In this paper we show that many directed graphs, like the perfect graphs, have stronger properties: for every maximal stable setS there exists a partition of the vertex set into paths which meet the stable set in only one point. Also: for every optimal coloring there exists a path which meets each color class in only one point. This suggests several conjecties similar to the perfect graph conjecture. Dedicated to Tibor Gallai on his seventieth birthday  相似文献   

18.
A ternary quasigroup (or 3-quasigroup) is a pair (N, q) where N is an n-set and q(x, y, z) is a ternary operation on N with unique solvability. A 3-quasigroup is called 2-idempotent if it satisfies the generalized idempotent law: q(x, x, y) = q(x, y, x) = q(y, x, x) = y. A conjugation of a 3-quasigroup, considered as an OA(3, 4, n), , is a permutation of the coordinate positions applied to the 4-tuples of . The subgroup of conjugations under which is invariant is called the conjugate invariant subgroup of . In this paper, we will complete the existence proof of the last undetermined infinite class of 2-idempotent 3-quasigroups of order n, n ≡ 1 (mod 4) and n > 9, with a conjugate invariant subgroup consisting of a single cycle of length four.   相似文献   

19.
Let Rbe a principal ideal ringRn the ring of n× nmatrices over R, and dk (A) the kth determinantal divisor of Afor 1 ? k? n, where Ais any element of Rn , It is shown that if A,BεRn , det(A) det(B:) ≠ 0, then dk (AB) ≡ 0 mod dk (A) dk (B). If in addition (det(A), det(B)) = 1, then it is also shown that dk (AB) = dk (A) dk (B). This provides a new proof of the multiplicativity of the Smith normal form for matrices with relatively prime determinants.  相似文献   

20.
A loop identity is of Bol-Moufang type if two of its three variables occur once on each side, the third variable occurs twice on each side, and the order in which the variables appear on both sides is the same, viz. ((xy)x)z = x(y(xz)). Loop varieties defined by one identity of Bol-Moufang type include groups, Bol loops, Moufang loops and C-loops. We show that there are exactly 14 such varieties, and determine all inclusions between them, providing all necessary counterexamples, too. This extends and completes the programme of Fenyves [Fe69]. Received October 23, 2003; accepted in final form April 12, 2005.  相似文献   

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