Idempotent Quasigroup Research Articles

Lagrange's theorem for Hom-Groups

Hom-groups are nonassociative generalizations of groups where the unitality and associativity are twisted by a map. We show that a Hom-group $(G, \alpha )$ is a pointed idempotent quasigroup (pique). We use Cayley tables of quasigroups to introduce some examples of Hom-groups. Introducing the notions of Hom-subgroups and cosets we prove Lagrange's theorem for finite Hom-groups. This states that the order of any Hom-subgroup $H$ of a finite Hom-group $G$ divides the order of $G$. We linearize Hom-groups to obtain a class of nonassociative Hopf algebras called Hom-Hopf algebras. As an application of our results, we show that the dimension of a Hom-sub-Hopf algebra of the finite dimensional Hom-group Hopf algebra $\mathbb {K}G$ divides the order of $G$. The new tools introduced in this paper could potentially have applications in theories of quasigroups, nonassociative Hopf algebras, Hom-type objects, combinatorics, and cryptography.

Study of Jordan quasigroups and their construction

ABSTRACTJordan quasigroups are commutative quasigroups satisfying the identity . In this paper we discuss the basic properties of Jordan quasigroups and prove that (i) every commutative idempotent quasigroup is Jordan quasigroup, (ii) if a Jordan quasigroup Q is distributive then Q is idempotent, (iii) an idempotent entropic quasigroup satisfies Jordan's identity, (iv) a unipotent quasigroup Q satisfying Jordan's identity satisfies left nuclear square property, (vi) if a quasigroup satisfies LC identity, then alternativity ⇔ Jordan's identity, (vii) for a unipotent Jordan quasigroup Q, and (viii) every Steiner quasigroup is Jordan quasigroup. We also prove some properties of the autotopism of Jordan loops. Moreover, we construct an infinite family of nonassociative Jordan quasigroups whose smallest member is of order 6.

An Affine Regular Icosahedron Inscribed in an Affine Regular Octahedron in a GS-Quasigroup

A golden section quasigroup or shortly a GS-quasigroup is an idempotent quasigroup which satises the identities a\dot (ab \dot c) \dot c = b; a\dot (a \dot bc) \dot c = b. The concept of a GS-quasigroup was introduced by VOLENEC. A number of geometric concepts can be introduced in a general GS-quasigroup by means of the binary quasigroup operation. In this paper, it is proved that for any affine regular octahedron there is an affine regular icosahedron which is inscribed in the given affine regular octahedron. This is proved by means of the identities and relations which are valid in a general GS-quasigrup. The geometrical presentation in the GS-quasigroup C(\frac{1}{2} (1 +\sqrt{5})) suggests how a geometrical consequence may be derived from the statements proven in a purely algebraic manner.

On the covering number of loops

A set of proper subgroups is a covering for a group if its union is the whole group. The minimal number of subgroups needed to cover G is called the covering number of G and is denoted by σ(G). It is an interesting problem in group theory to determine integers n such that there exists a group G with σ(G)=n and n−1>1 is not a power of a prime. It is known that there exist integers n>2, which are not covering numbers of a group. So far it has been shown that for n<27 the integers 7, 11, 19, 21, 22 and 25 are not covering numbers of a group. It is an open question if the set of integers with this property is finite or infinite. In this paper an idempotent quasigroup of order n>2 with the property that every two distinct elements generate the entire quasigroup is obtained. With this it is shown that for any integer n>2 there exists a loop whose covering number is n.

Large sets of idempotent quasigroups with resolvability

An idempotent quasigroup (IQ) possessing a unipotent orthogonal mate is called resolvable. An idempotent quasigroup possessing an idempotent orthogonal mate is called almost resolvable. A large set of idempotent quasigroups of order v is a partition of all 3-vectors with 3 distinct components of a <italic>v</italic>-set into <italic>v</italic>-2 IQ(<italic>v</italic>)s pairwise agreeing on only the idempotent rows. In this paper, large sets of resolvable idempotent quasi-groups, large sets of almost resolvable idempotent quasigroups, and large sets of resolvable symmetric idempotent quasigroups (resolvable golf designs) are investigated by utilizing t-balanced designs with <italic>t</italic>=2, 3.

Overlarge sets of resolvable idempotent quasigroups

An idempotent quasigroup (X,∘) of order v is called resolvable (denoted by RIQ(v)) if the set of v(v−1) non-idempotent 3-vectors {(a,b,a∘b):a,b∈X,a≠b} can be partitioned into v−1 disjoint transversals. An overlarge set of idempotent quasigroups of order v, briefly by OLIQ(v), is a collection of v+1 IQ(v)s, with all the non-idempotent 3-vectors partitioning all those on a (v+1)-set. An OLRIQ(v) is an OLIQ(v) with each member IQ(v) being resolvable. In this paper, it is established that there exists an OLRIQ(v) for any positive integer v≥3, except for v=6, and except possibly for v∈{10,11,14,18,19,23,26,30,51}. An OLIQ♢(v) is another type of restricted OLIQ(v) in which each member IQ(v) has an idempotent orthogonal mate. It is shown that an OLIQ♢(v) exists for any positive integer v≥4, except for v=6, and except possibly for v∈{14,15,19,23,26,27,30}.

A graph-theoretic approach to quasigroup cycle numbers

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.

Transitive resolvable idempotent quasigroups and large sets of resolvable Mendelsohn triple systems

We first define a transitive resolvable idempotent quasigroup (TRIQ), and show that a TRIQ of order v exists if and only if 3 ∣ v and v ⁄ ≡ 2 ( mod 4 ) . Then we use TRIQ to present a tripling construction for large sets of resolvable Mendelsohn triple systems LRMTS ( v ) s, which improves an earlier version of tripling construction by Kang. As an application we obtain an LRMTS ( 4 ⋅ 3 n ) for any integer n ≥ 1 , which provides an infinite family of even orders.

Tripling Construction for Large Sets of Resolvable Directed Triple Systems

In this paper, we first define a doubly transitive resolvable idempotent quasigroup (DTRIQ), and show that a DTRIQ of order v exists if and only if v ≡ 0 (mod 3) and v ≢ 2 (mod 4). Then we use DTRIQ to present a tripling construction for large sets of resolvable directed triple systems, which improves an earlier version of tripling construction by Kang (J. Combin. Designs, 4 (1996), 301–321). As an application, we obtain an LRDTS(4 · 3 n ) for any integer n ≥ 1, which provides an infinite family of even orders.

THE EXISTENCE OF OVERLARGE SETS OF IDEMPOTENT QUASIGROUPS

A idempotent quasigroup (Q, o) of order n is equivalent to an n(n – 1) × 3 partial orthogonal array in which all of rows consist of 3 distinct elements. Let X be a (n + 1)-set. Denote by T(n + 1) the set of (n + 1)n(n – 1) ordered triples of X with the property that the 3 coordinates of each ordered triple are distinct. An overlarge set of idempotent quasigroups of order n is a partition of T(n + 1) into n + 1 n(n – 1) × 3 partial orthogonal arrays Ax, x ɛ X based on X \\ {x}. This article gives an almost complete solution of overlarge sets of idempotent quasigroups.

The existence of Schröder designs with equal-sized holes

A holey Schröder design of type h1n1h2n2 … hknk (HSD(h1n1h2n2 … hknk)) is equivalent to a frame idempotent Schröder quasigroup (FISQ(h1n1h2n2 … hknk)) of order n with ni missing subquasigroups (holes) of order hi, (1 ⩽ i ⩽ k), which are disjoint and spanning, that is, Σ1 ⩽ i ⩽ knihi = n. In this paper, it is shown that an HSD(hn) exists if and only if h2n(n − 1) 0 (mod 4) with expceptions (h, n) ϵ {{(1,5),(1,9),(2,4)}} and the possible exception of (h, n) = (6,4).

Incomplete idempotent Schröder quasigroups and related packing designs

In this paper it is proved that, for any positive integern ≡ 2, 3 (mod 4),n ≥ 7, there exists an incomplete idempotent Schroder quasigroup with one hole of size two IISQ(n, 2) except forn = 10. It is also proved that for any positive integern ≡ 0, 1 (mod 4), there exists an idempotent Schroder quasigroup ISQ(n) except forn = 5 and 9. These results completely determine the spectrum of ISQ(n) and provide an application to the packing of a class of edge-coloured block designs.