The spectrum of 2‐idempotent 3‐quasigroups with conjugate invariant subgroups

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     

On the invariant spectrum on

Motivated by the work of Abreu and Freitas 1 , we study the invariant spectrum of the Laplace operator associated to hermitian line bundles endowed with invariant metrics over .     

2-idempotent 3-quasigroups with a conjugate invariant subgroup consisting of a single cycle of length four

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.      

On the conjugate type vector and the structure of a normal subgroup

Czechoslovak Mathematical Journal - Let N be a normal subgroup of a group G. The structure of N is given when the G-conjugacy class sizes of N is a set of a special kind. In fact, we give the...     

On the construction of cyclic quasigroups

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 w_i+2(x,y) = w_i(x,y)w_i+1(x,y). An identity of the form w_3n(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.     

On quasigroups with few associative triples

There are numerous application of quasigroups in cryptology. It turns out that quasigroups with the relatively small number of associative triples can be utilized in designs of hash functions. In this paper we provide both a new lower bound and a new upper bound on the minimum number of associative triples over quasigroups of a given order.     

The spectrum for quasigroups with cyclic automorphisms and additional symmetries

We determine necessary and sufficient conditions for the existence of a quasigroup of order n having an automorphism consisting of a single cycle of length m and n−m fixed points, and having any combination of the additional properties of being idempotent, unipotent, commutative, semi-symmetric or totally symmetric. Quasigroups with such additional properties and symmetries are equivalent to various classes of triple systems.     

The spectrum for large sets of idempotent quasigroups

In this article we construct a large set of idempotent quasigroups of order 14. Combined with the results in Chang, JCMCC; and Teirlinck and Lindner, Eur J Combin 9 (1988), 83–89, this shows that the spectrum for large sets of idempotent quasigroups of order n [briefly, LIQ(n)] is the set all integers n≥ 3 with the exception of n=6. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 79–82, 2000     

On linear representations of quasigroups

For homotopies of quasigroups, an analog of the fundamental theorem on homomorphisms does not hold in general. In this paper, we consider two approaches that allow one to obtain an analog of this theorem: the introduction of strict homotopies and the move from quasigroups to three-sorted quasigroups.     

On congruences of groupoids closely connected with quasigroups

Conditions when a congruence of a left (right) division groupoid and a left (right) cancellation groupoid is closed (“normal”) are given. Conditions for the simplicity of the above-mentioned groupoids are obtained.     

On algebras of the Temperley-Lieb type associated with algebras generated by generators with given spectrum

We introduce and study algebras of the Temperley—Lieb type associated with algebras generated by linearly connected generators with given spectrum. We study their representations and the sets of parameters for which representations of these algebras exist.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 634–641, May, 2004.     

On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

On the classification of functional equations on quasigroups

We consider functional equations over quasigroup operations. We prove that every quadratic parastrophically uncancelable functional equation for four object variables is parastrophically equivalent to the functional equation of mediality or the functional equation of pseudomediality. The set of all solutions of the general functional equation of pseudomediality is found and a criterion for the uncancelability of a quadratic functional equation for four object variables is established.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1259–1266, September, 2004.