ALGEBRAIC REPRESENTATION OF A CLASS OF HOMOGENEOUS STEINER QUADRUPLE SYSTEMS

Abstract

A Steiner system \(S(t,k,v)\) is a pair \((Q,B)\) of \(v\)-element set \(Q\) and a collection \(B\) of its \(k\)-element subsets (blocks), such that every \(t\)-element subset of \(Q\) is contained in exactly one block. Systems \(S(2,3,v)\) are Steiner triple systems (\(STS\)) and their algebraic representatives are the idempotent totally symmetric quasigroups. Steiner quadruple systems (\(SQS\)) are systems \(S(3,4,v)\), represented by the idempotent totally symmetric ternary quasigroups. For \(SQS\) \((Q,B)\) and \(a\in Q\), by taking the set \(Q\setminus\{a\}\) and the blocks \(\{\{x,y,z\}\mid\{x,y,z,a\}\in B\}\), a derived triple system is obtained. An \(SQS\) is called homogenous if all of its derived triple systems are isomorphic. In this paper sufficient conditions for \(SQS\) to be homogenous are given, resulting with an algebraic representation of one class of homogenous quadruple systems.