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1.
In this paper we prove that a finite partial commutative (idempotent commutative) Latin square can be embedded in a finite commutative (idempotent commutative) Latin square. These results are then used to show that the loop varieties defined by any non-empty subset of the identities {x(xy) = y, (yx)x = y} and the quasi-group varieties defined by any non-empty subset of {x2 = x, x(xy) = y, (yx)x = y}, except possibly {x(xy) = y, (yx)x = y}, have the strong finite embeddability property. It is then shown that the finitely presented algebras in these varities are residually finite, Hopfian, and have a solvable word problem. 相似文献
2.
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
3.
Diane Johnson 《Discrete Mathematics》1977,19(3):265-271
The variety of groupoids defined by the identitites (yx)x = xy and ((xy)(yx))(xy) = y has the properties that every groupoid generated by two elements is of order 11. The two generating identities imply others with a wide variety of combinatorial implications. 相似文献
4.
J. Barnhardt 《Aequationes Mathematicae》1978,18(1-2):304-321
The structure of all distributive topological groupoidsM on a closed or half-closed real interval is completely determined provided one endpoint is a zero forM, M contains an injective idempotent e, and (ex)(ye)= (ey)(xe) holds for allx, y inM. A groupoid is distributive if (xy)z = (xz)(yz) andx(yz)= (xy)(xz) hold for allx, y, z inM. An idempotente is called injective ifx =y wheneverxe =ye orex =ey. It is a corollary that all such groupoids are actually medial. The proof is accomplished by showing that, in a certain sense, distributivity corresponds to biassociativity as mediality corresponds to associativity. A groupoid is called biassociative it the subgroupoid generated by each pair of elements in the groupoid is associative. 相似文献
5.
6.
Vedran Krčadinac 《Mathematica Slovaca》2011,61(6):885-894
Napoleon’s quasigroups are idempotent medial quasigroups satisfying the identity (ab·b)(b·ba) = b. In works by V. Volenec geometric terminology has been introduced in medial quasigroups, enabling proofs of many theorems
of plane geometry to be carried out by formal calculations in a quasigroup. This class of quasigroups is particularly suited
for proving Napoleon’s theorem and other similar theorems about equilateral triangles and centroids. 相似文献
7.
F.E Bennett 《Journal of Combinatorial Theory, Series A》1982,33(1):117-119
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
8.
A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x
1, ...,x
n
),x
1, ...,x
n–1) =x
n
is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated. 相似文献
9.
Charles C. Lindner 《Discrete Mathematics》1973,6(2):149-158
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
10.
The idempotent graph of a ring R, denoted by I(R), is a graph whose vertices are all nontrivial idempotents of R and two distinct vertices x and y are adjacent if and only if xy = yx = 0. In this paper, we show that diam\({(I(M_n(D))) = 4}\), for all natural number \({n \geq 4}\) and diam\({(I(M_3(D))) = 5}\), where D is a division ring. We also provide some classes of rings whose idempotent graphs are connected. Moreover, the regularity, clique number and chromatic number of idempotent graphs are studied. 相似文献
11.
12.
Extending the notions of inverse transversal and associate subgroup, we consider a regular semigroup S with the property that there exists a subsemigroup T which contains, for each x∈S, a unique y such that both xy and yx are idempotent. Such a subsemigroup is necessarily a group which we call a special subgroup. Here, we investigate regular semigroups with this property. In particular, we determine when the subset of perfect elements is a subsemigroup and describe its structure in naturally arising situations. 相似文献
13.
Sherman Stein 《Algebra Universalis》2014,71(4):359-373
Tamura proved that for any semigroup word U(x, y), if every group satisfying an identity of the form yx ~ xU(x, y)y is abelian, then so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show that a group is abelian than that a semigroup is. If we know that it is, then there must be a sequence of substitutions using xU(x, y)y ~ yx that transforms xy to yx. We examine such sequences and propose finding them as a challenge to proof by computer. Also, every model of y ~ xU(x, y)x is a group. This raises a similar challenge, which we explore in the special case y ~ x m y p x n . In addition, we determine the free model with two generators of some of these identities. In particular, we find that the free model for y ~ x 2 yx 2 has order 32 and is the product of D 4 (the symmetries of a square), C 2, and C 4, and point out relations between such identities and Burnside’s Problem concerning models of x n ~ y n . We also examine several identities not related to groups. 相似文献
14.
15.
J.W Moon 《Journal of Combinatorial Theory, Series B》1976,21(1):71-75
An arc in a tournament is bad if there exists no path of length two from x to y. Formulas are found for the number of tournaments Tn whose bad arcs determine a spanning cycle or path. 相似文献
16.
Dilian Yang 《Aequationes Mathematicae》2011,82(3):299-318
As a continuation of An and Yang (Integral Equ Oper Theory 66:183–195, 2010) in this paper, the symmetrized Sine addition formula
w(xy)+w(yx)=2f(x)w(y)+2w(x)f(y) w(xy)+w(yx)=2f(x)w(y)+2w(x)f(y) 相似文献
17.
Ali Reza Moghaddamfar 《Siberian Mathematical Journal》2006,47(5):911-914
The noncommuting graph ?(G) of a nonabelian finite group G is defined as follows: The vertices of ?(G) are represented by the noncentral elements of G, and two distinct vertices x and y are joined by an edge if xy ≠ yx. In [1], the following was conjectured: Let G and H be two nonabelian finite groups such that ?(G) ? ?(H); then ¦G¦ = ¦H¦. Here we give some counterexamples to this conjecture. 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), $({{N}},{\mathcal{B}})$, is a permutation of the coordinate positions applied to the 4‐tuples of ${\mathcal{B}}$. The subgroup of conjugations under which $({{N}},{\mathcal{B}})$ is invariant is called the conjugate invariant subgroup of $({{N}},{\mathcal{B}})$. In this article, we determined the existence of 2‐idempotent 3‐quasigroups of order n, n≡7 or 11 (mod 12) and n≥11, with conjugate invariant subgroup consisting of a single cycle of length three. This result completely determined the spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups. As a corollary, we proved that an overlarge set of Mendelsohn triple system of order n exists if and only if n≡0, 1 (mod 3) and n≠6. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 292–304, 2010 相似文献
19.
Brent Kerby 《Journal of Combinatorial Theory, Series A》2011,118(8):2232-2245
Norton and Stein associated a number with each idempotent quasigroup or diagonalized Latin square of given finite order n, showing that it is congruent mod 2 to the triangular number T(n). In this paper, we use a graph-theoretic approach to extend their invariant to an arbitrary finite quasigroup. We call it the cycle number, and identify it as the number of connected components in a certain graph, the cycle graph. The congruence obtained by Norton and Stein extends to the general case, giving a simplified proof (with topology replacing case analysis) of the well-known congruence restriction on the possible orders of general Schroeder quasigroups. Cycle numbers correlate nicely with algebraic properties of quasigroups. Certain well-known classes of quasigroups, such as Schroeder quasigroups and commutative Moufang loops, are shown to maximize the cycle number among all quasigroups belonging to a more general class. 相似文献
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