Steiner systems and \(n^{-1}\)-isomorphisms
Let \(k\) and \(v\) be positive integers, \(k > 2\), \(v > 2\). A partial system \(S(2,k,v)\) is an incidence structure \((\mathcal V, \mathcal T)\) where \(\mathcal V\) is a set of \(v\) points, \(\mathcal T\) is the set of blocks, each block contains exactly \(k\) points and for every unordered pair of distinct points there is at most one block containing them. A partial system \(S(2,k,v)\) is called a Steiner system if for every unordered pair of distinct points there is exactly one block containing them. In this paper the authors introduce the concept of \(n^{-1}\)-isomorphism between Steiner systems. This coincides with the concept of isomorphism whenever \(n = 1\). Two Steiner systems \(S_1\) and \(S_2\) are said to be \(n^{-1}\)-isomorphic if there exist \(n\) partial systems \(S_i^{(1)},\cdots,S_i^{(n)}\) contained in \(S_i\), \(i\in \{1,2\}\), such that \(S_1^{(k)}\) and \(S_2^{(k)}\) are isomorphic for each \(k\in \{1,\cdots,n\}\). The \(n^{-1}\)-isomorphisms are used to study nets replacements [see \textit{T. G. Ostrom}, ``Finite translation planes'', Springer Verlag (1970; Zbl 0205.49901)], and to study the transformation methods of designs and other incidence structures introduced by \textit{G. Quattrocchi} and \textit{L. A. Rosati} [Geom. Dedicata 44, No. 2, 233-240 (1992; Zbl 0773.51007)] and generalized by \textit{G. Rinaldi} [J. Geom. 48, No. 1-2, 167-173 (1993; Zbl 0788.51001)].
scientific article; zbMATH DE number 1000354