Triple systems and binary operations

Pacific Journal of Mathematics, 1970

Graphs and Combinatorics, 2010

A Steiner triple system of order v, denoted ST S(v), is said to be tricyclic if it admits an automorphism whose disjoint cyclic decomposition consists of three cycles. In this paper we give necessary and sufficient conditions for the existence of a tricyclic ST S(v) for several cases. We also pose conjectures concerning their existence in two remaining cases. Keywords Steiner triple systems • Tricyclic automorphism 1 Introduction A Steiner triple system of order v, denoted ST S(v), is a v-element set, X , of points, together with a set β, of unordered triples of elements of X , called blocks, such that any two points of X are together in exactly one block of β. It is well known that a ST S(v) exists if and only if v ≡ 1 or 3 (mod 6). For a general review of triple systems in general, see [5]. An automorphism of a ST S is a permutation π of X which fixes β. A permutation π of a v-element set is said to be of type [π ] = [π 1 , π 2 ,. .. , π v

Discrete Mathematics, 1973

Journal of Algebra and Its Applications, 2015

The paper is devoted to the study of free objects in the variety of Steiner loops and of the combinatorial structures behind them, focusing on their automorphism groups.

A threefold triple system is a design wherein each pair of treatments occurs exactly once. One way to construct this design is by using an idempotent commutative quasigroup. This paper attempts to provide another method of constructing a 3-fold triple system. Firstly, we would like to discuss compatible factorization without multiple edges using a patterned starter construction. Then, we will use this construction to enumerate a distinct 3-fold triple system for every odd order v > 3.

FUDMA JOURNAL OF SCIENCES

In this paper, we have examine some properties of elements of the semigroup , where DX, is the set of all binary relations α ⊆ X × X satisfying , (), and is a binary operation on DX defined by () , with xα denoting set of images of x under α, and yβ−1 denoting set of pre-images of y under β. In particular, we showed that in the semigroup there is no distinction between the concepts of reflexive and symmetric relations. We also presented a characterization of idempotent elements in in term of equivalence relations.

Australas. J Comb., 1995

An extended triple system with no idempotent element (ETS) is a collection of non ordered triples of type {x,y,z} or {x,x,y} chosen from a v -set in such a way that each pair (whether distinct or not) is contained in exactly one triple. (For example, in the block {x,x,y}, the pair {x,y} is said to occur one tinle.) Such a design has "'v = v(v + 3) /6 blocks and a necessary and sufficient condition for existence is that v ° (mod 3). Let J( v) denote the set of non -negative integers k such that there exist two ETS(v) with precisely k blocks in common. In this paper we determine J( v) for all admissible v, in particular we show that J( ) 1( )-{13} and J(v) =1(v), where l{v) ={O,l, ... , sv-3, s~} The concept of an extended triple system was introduced by D.M. Johnson and N.S. Mendelsohn . An extended triple system is a pair (V,B), where V is a finite set and B is a collection of non -ordered triples from V , where each triple may have repeated elements, such that every pair of elements of V, not necessarily distinct, is contained in exactly one triple of B. The triple of B are of three types (1) {x,x,:r} , (2) {y,y,z} and (3) {a,b,c} , where the element x is called an idempotent and y a non -idempotent of the system (V,B). We shall denote by {v ; a} the class of all extended triple systems on v -elements containing exactly 0: idempotent elements.

arXiv: Group Theory, 2017

In this paper we introduce novel views of monoids and groups. More specifically, for a given set $S$, let $S^{S\times S}$ be the set of binary operations on $S$. We equip $S^{S\times S}$ with canonical binary operations induced by the elements of $S$. Let $S^{S\times S}_{mn}$ (respectively, $S^{S\times S}_{gr}$) be the set of binary operations that make $S$ monoids (respectively, groups). Then we have the following "duality": for each $z\in S^{S\times S}_{mn}$ a certain subset of $S^{S\times S}$, denoted by $S^*_z$, is a monoid with a canonical binary operation and is isomorphic to $(S,z)$. If $z\in S^{S\times S}_{gr}$, then $S^{S\times S}_{gr}$ can be partitioned into copies of $S^*_z$. We also give a new characterization of group binary operations which distinguishes them from the other binary operations. These results give us new insights into monoids and groups, and will provide new tools and directions in studying these objects.

Mathematische Zeitschrift, 1974

Proceedings of the London Mathematical Society, 1980

We prove that a partial Steiner triple system 8 of order n can be embedded in a Steiner triple system T of any given admissible order greater than 4w. Furthermore, if G(S), the missing-edge graph of S, has the property that A(G)<ri(n + l)l and \E(G)\ then # can be embedded in a Steiner triple system of order 2n +1, provided that 2w +1 is admissible.