On the structure of split Leibniz triple systems

We study the structure of arbitrary split Leibniz triple systems with a coherent 0-root space. By developing techniques of connections of roots for this kind of triple systems, under certain conditions, in the case of T being of maximal length, the simplicity of the Leibniz triple systems is characterized.     

Dialgebras and related triple systems

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz-Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz-Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an (ε, δ)-Freudenthal-Kantor triple system on a dialgebra.     

Some new perfect Steiner triple systems

In a Steiner triple system STS(v) = (V, B), for each pair {a, b} ⊂ V, the cycle graph G_a,b can be defined as follows. The vertices of G_a,b are V \ {a, b, c} where {a, b, c} ∈ B. {x, y} is an edge if either {a, x, y} or {b, x, y} ∈ B. The Steiner triple system is said to be perfect if the cycle graph of every pair is a single (v − 3)-cycle. Perfect STS(v) are known only for v = 7, 9, 25, and 33. We construct perfect STS (v) for v = 79, 139, 367, 811, 1531, 25771, 50923, 61339, and 69991. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 327–330, 1999     

On split Lie triple systems

We begin the study of arbitrary split Lie triple systems by focussing on those with a coherent 0-root space. We show that any such triple systems T with a symmetric root system is of the form T = + Μ _j I _j with a subspace of the 0-root space T ₀ and any I _j a well described ideal of T, satisfying [I _j , T, I _k ] = 0 if j ≠ k. Under certain conditions, it is shown that T is the direct sum of the family of its minimal ideals, each one being a simple split Lie triple system, and the simplicity of T is characterized. The key tool in this job is the notion of connection of roots in the framework of split Lie triple systems.     

Classification of a subclass of low-dimensional complex filiform Leibniz algebras

We give a complete classification of a subclass of complex filiform Leibniz algebras obtained from naturally graded non-Lie filiform Leibniz algebras. The isomorphism criteria in terms of invariant functions are given.     

On Integrable roots in split Lie triple systems

We focus on the notion of an integrable root in the framework of split Lie triple systems T with a coherent 0-root space. As a main result, it is shown that if T has all its nonzero roots integrable, then its standard embedding is a split Lie algebra having all its nonzero roots integrable. As a consequence, a local finiteness theorem for split Lie triple systems, saying that whenever all nonzero roots of T are integrable then T is locally finite, is stated. Finally, a classification theorem for split simple Lie triple systems having all its nonzero roots integrable is given.     

Steiner triple systems with two disjoint subsystems

It is shown that there exists a triangle decomposition of the graph obtained from the complete graph of order v by removing the edges of two vertex disjoint complete subgraphs of orders u and w if and only if u,w, and v are odd, (mod 3), and . Such decompositions are equivalent to group divisible designs with block size 3, one group of size u, one group of size w, and v – u – w groups of size 1. This result settles the existence problem for Steiner triple systems having two disjoint specified subsystems, thereby generalizing the well‐known theorem of Doyen and Wilson on the existence of Steiner triple systems with a single specified subsystem. © 2005 Wiley Periodicals, Inc. J Combin Designs     

Sequencing partial Steiner triple systems

A partial Steiner triple system of order $n$ is sequenceable if there is a sequence of length $n$ of its distinct points such that no proper segment of the sequence is a union of point‐disjoint blocks. We prove that if a partial Steiner triple system has at most three point‐disjoint blocks, then it is sequenceable.     

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 inside outside intersection problem for hexagon triple systems

We give a complete solution of the following two problems:

1. (1)
   For which (n, x) does there exist a pair of hexagon triple systems of order n having x inside triples in common?
    
2. (2)
   For which (n, x) does there exist a pair of hexagon triple systems having x outside triples in common?
    

     

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 .

     

Leibniz代数胚上动力系统的轨道范例与图示

首先陈述Leibniz代数胚上的动力系统和在局部坐标系下该动力系统的方程,在此基础上,给出了动力系统轨道的范例和图示.     

Embedding hypertrees into steiner triple systems

In this paper, we are interested in the following question: given an arbitrary Steiner triple system on vertices and any 3‐uniform hypertree on vertices, is it necessary that contains as a subgraph provided ? We show the answer is positive for a class of hypertrees and conjecture that the answer is always positive.     

On 6-sparse Steiner triple systems

We give the first known examples of 6-sparse Steiner triple systems by constructing 29 such systems in the residue class 7 modulo 12, with orders ranging from 139 to 4447. We then present a recursive construction which establishes the existence of 6-sparse systems for an infinite set of orders. Observations are also made concerning existing construction methods for perfect Steiner triple systems, and we give a further example of such a system. This has order 135,859 and is only the fourteenth known. Finally, we present a uniform Steiner triple system of order 180,907.     

Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a,b,c}, {c,d,e}, {e,f,a}, {x,a,y}, {x,c,z}, {x,e,w}, where a,b,c,d,e,f,x,y,z and w are distinct, is called a dexagon triple. In this case the six edges {a,c}, {c,e}, {e,a}, {x,a}, {x,c}, and {x,e} form a copy of K₄ and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X,D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K_v. A dexagon triple system is said to be perfect if the inside copies of K₄ form a block design. In this note, we investigate the existence of a dexagon triple system with a subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient.     

Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3-cycles (or triples) (a,b,c), (c,d,e), and (e,f,a), where a,b,c,d,e and f are distinct is called a hexagon triple. The 3-cycle (a,c,e) is called an inside 3-cycle; and the 3-cycles (a,b,c), (c,d,e), and (e,f,a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X,C), where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3K_v. Note that the outside 3-cycles form a 3-fold triple system. If the hexagon triple system has the additional property that the collection of inside 3-cycles (a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order v if and only if and v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order v with a perfect sub-hexagon triple system of order u if and only if v≥2u+1, and u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.