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.     

Completing partial commutative quasigroups constructed from partial Steiner triple systems is NP-complete

Deciding whether an arbitrary partial commutative quasigroup can be completed is known to be NP-complete. Here, we prove that it remains NP-complete even if the partial quasigroup is constructed, in the standard way, from a partial Steiner triple system. This answers a question raised by Rosa in [A. Rosa, On a class of completable partial edge-colourings, Discrete Appl. Math. 35 (1992) 293-299]. To obtain this result, we prove necessary and sufficient conditions for the existence of a partial Steiner triple system of odd order having a leave L such that E(L)=E(G) where G is any given graph.     

Closures of totally geodesic immersions into locally symmetric spaces of noncompact type

It is established that if and are connected locally symmetric spaces of noncompact type where has finite volume, and is a totally geodesic immersion, then the closure of in is an immersed ``algebraic' submanifold. It is also shown that if in addition, the real ranks of and are equal, then the the closure of in is a totally geodesic submanifold of The proof is a straightforward application of Ratner's Theorem combined with the structure theory of symmetric spaces.

     

Embedding 5-cycle systems into pentagon triple systems

We show that the spectrum for pentagon triple systems is the set of all n≡1,15,21 or . We then construct a 5-cycle system of order 10n+1 which can be embedded in a pentagon triple system of order 30n+1 and also construct a 5-cycle system of order 10n+5 which can be embedded in a pentagon triple system of order 30n+15, with the possible exception of embedding a 5-cycle system of order 21 in a pentagon triple system of order 61.     

The parity of the number of quasigroups

Let I(n) be the number of isomorphism classes of quasigroups of order n. Despite prior enumerations showing that I(n) is odd for 1≤n≤11, we find that I(12) is even. We also give a method for finding the parity of I(n), which we use to show that I(n) is odd for n∈{13,14,15,16,17,19,21}.     

The existence spectrum of golf designs

In this note, a golf design of order 41 is constructed. Combined Colbourn and Nonay's result, the existence spectrum of golf design of order υ is the set {υ: υ≡1 (mod 2), υ ≥ 3, υ ≠ 5}. © 2005 Wiley Periodicals, Inc. J Combin Designs 15: 84–89, 2007     

The recognition of symmetric latin squares

A latin square S is isotopic to another latin square S′ if S′ can be obtained from S by permuting the row indices, the column indices and the symbols in S. Because the three permutations used above may all be different, a latin square which is isotopic to a symmetric latin square need not be symmetric. We call the problem of determining whether a latin square is isotopic to a symmetric latin square the symmetry recognition problem. It is the purpose of this article to give a solution to this problem. For this purpose we will introduce a cocycle corresponding to a latin square which transforms very simply under isotopy, and we show this cocycle contains all the information needed to determine whether a latin square is isotopic to a symmetric latin square. Our results relate to 1‐factorizations of the complete graph on n + 1 vertices, K_{n + 1}. There is a well known construction which can be used to make an n × n latin square from a 1‐factorization on n + 1 vertices. The symmetric idempotent latin squares are exactly the latin squares that result from this construction. The idempotent recognition problem is simple for symmetric latin squares, so our results enable us to recognize exactly which latin squares arise from 1‐factorizations of K_{n + 1}. As an example we show that the patterned starter 1‐factorization for the group G gives rise to a latin square which is in the main class of the Cayley latin square for G if and only if G is abelian. Hence, every non‐abelian group gives rise to two latin squares in different main classes. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 291–300, 2008     

Embedding Steiner triple systems in hexagon triple systems

A hexagon triple is the graph consisting of the three triangles (triples) {a,b,c},{c,d,e}, and {e,f,a}, where a,b,c,d,e, and f are distinct. The triple {a,c,e} is called an inside triple. A hexagon triple system of order n is a pair (X,H) where H is a collection of edge disjoint hexagon triples which partitions the edge set of K_n with vertex set X. The inside triples form a partial Steiner triple system. We show that any Steiner triple system of order n can be embedded in the inside triples of a hexagon triple system of order approximately 3n.     

Embedding partial Steiner triple systems so that their automorphisms extend

It is shown that there is a function g on the natural numbers such that a partial Steiner triple system U on u points can be embedded in a Steiner triple system V on ν points, in such a way that all automorphisms of U can be extended to V, for every admissible ν satisfying ν > g(u). We find exponential upper and lower bounds for g. © 2005 Wiley Periodicals, Inc. J Combin Designs.     

On semi-planar Steiner quasigroups

A Steiner triple system (briefly ST) is in 1-1 correspondence with a Steiner quasigroup or squag (briefly SQ) [B. Ganter, H. Werner, Co-ordinatizing Steiner systems, Ann. Discrete Math. 7 (1980) 3-24; C.C. Lindner, A. Rosa, Steiner quadruple systems: A survey, Discrete Math. 21 (1979) 147-181]. It is well known that for each n≡1 or 3 (mod 6) there is a planar squag of cardinality n [J. Doyen, Sur la structure de certains systems triples de Steiner, Math. Z. 111 (1969) 289-300]. Quackenbush expected that there should also be semi-planar squags [R.W. Quackenbush, Varieties of Steiner loops and Steiner quasigroups, Canad. J. Math. 28 (1976) 1187-1198]. A simple squag is semi-planar if every triangle either generates the whole squag or the 9-element squag. The first author has constructed a semi-planar squag of cardinality 3n for all n>3 and n≡1 or 3 (mod 6) [M.H. Armanious, Semi-planar Steiner quasigroups of cardinality 3n, Australas. J. Combin. 27 (2003) 13-27]. In fact, this construction supplies us with semi-planar squags having only nontrivial subsquags of cardinality 9. Our aim in this article is to give a recursive construction as n→3n for semi-planar squags. This construction permits us to construct semi-planar squags having nontrivial subsquags of cardinality >9. Consequently, we may say that there are semi-planar (or semi-planar ) for each positive integer m and each n≡1 or 3 (mod 6) with n>3 having only medial subsquags at most of cardinality 3^ν (sub-) for each ν∈{1,2,…,m+1}.     

New families of atomic Latin squares and perfect 1-factorisations

A perfect 1-factorisation of a graph G is a decomposition of G into edge disjoint 1-factors such that the union of any two of the factors is a Hamiltonian cycle. Let p?11 be prime. We demonstrate the existence of two non-isomorphic perfect 1-factorisations of K_p+1 (one of which is well known) and five non-isomorphic perfect 1-factorisations of K_p,p. If 2 is a primitive root modulo p, then we show the existence of 11 non-isomorphic perfect 1-factorisations of K_p,p and 5 main classes of atomic Latin squares of order p. Only three of these main classes were previously known. One of the two new main classes has a trivial autotopy group.     

Self‐embeddings of cyclic and projective Steiner quasigroups

It is shown that for every admissible order v for which a cyclic Steiner triple system exists, there exists a biembedding of a cyclic Steiner quasigroup of order v with a copy of itself. Furthermore, it is shown that for each n≥2 the projective Steiner quasigroup of order 2ⁿ?1 has a biembedding with a copy of itself. © 2010 Wiley Periodicals, Inc. J Combin Designs 19:16‐27, 2010     

Extended directed triple systems

Let {n;b₂,b₁} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b₂ and the number of blocks of the form [b,a,a] or [a,a,b] is b₁. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b₂,b₁} is b₁≠1, 0?b₂+b₁?n and

(1)

for ;

(2)

for .

     

Symmetric subvarieties in compactifications and the Radon transform on Riemannian symmetric spaces of noncompact type

We investigate the totally geodesic Radon transform which assigns a function to its integration over totally geodesic symmetric submanifolds in Riemannian symmetric spaces of noncompact type. Our main concern is focused on the injectivity and support theorem. Our approach is based on the projection slice theorem relating the totally geodesic Radon transform and the Fourier transforms on symmetric spaces. Our approach also uses the study of geometry concerned with the totally geodesic symmetric subvarieties in Riemannian symmetric spaces in terms of the cell structure of the Satake compactifications.     

Large sets of extended directed triple systems with odd orders

For three types of triples, unordered, cyclic and transitive, the corresponding extended triple, extended triple system and their large set are introduced. The spectrum of LEDTS(v) for even v has been given in our paper (Liu and Kang (2009) [9]). In this paper, we shall discuss the existence problem of LEDTS(v) for odd v and give the almost complete conclusion: there exists an LEDTS(v) for any positive integer v≠4 except possible v=95,143,167,203,215.     

Algebraic properties of quantum quasigroups

Quantum quasigroups provide a self-dual framework for the unification of quasigroups and Hopf algebras. This paper furthers the transfer program, investigating extensions to quantum quasigroups of various algebraic features of quasigroups and Hopf algebras. Part of the difficulty of the transfer program is the fact that there is no standard model-theoretic procedure for accommodating the coalgebraic aspects of quantum quasigroups. The linear quantum quasigroups, which live in categories of modules under the direct sum, are a notable exception. They form one of the central themes of the paper.From the theory of Hopf algebras, we transfer the study of grouplike and setlike elements, which form separate concepts in quantum quasigroups. From quasigroups, we transfer the study of conjugate quasigroups, which reflect the triality symmetry of the language of quasigroups. In particular, we construct conjugates of cocommutative Hopf algebras. Semisymmetry, Mendelsohn, and distributivity properties are formulated for quantum quasigroups. We classify distributive linear quantum quasigroups that furnish solutions to the quantum Yang-Baxter equation. The transfer of semisymmetry is designed to prepare for a quantization of web geometry.     

The existence of large set of symmetric partitioned incomplete latin squares

In this paper, we investigate the existence of large sets of symmetric partitioned incomplete latin squares of type g^u (LSSPILSs) which can be viewed as a generalization of the well‐known golf designs. Constructions for LSSPILSs are presented from some other large sets, such as golf designs, large sets of group divisible designs, and large sets of Room frames. We prove that there exists an LSSPILS(g^u) if and only if u ≥ 3, g(u ? 1) ≡ 0 (mod 2), and (g, u) ≠ (1, 5).