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The structure of nilpotent steiner quadruple systems

✍ Scribed by Andreas J. Guelzow

Publisher
John Wiley and Sons
Year
1993
Tongue
English
Weight
796 KB
Volume
1
Category
Article
ISSN
1063-8539

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✦ Synopsis

Steiner quadruple systems can be coordinatized by SQS-skeins. We investigate those Steiner quadruple systems that correspond to finite nilpotent SQS-skeins. S. Klossek has given representation and construction theorems for finite distributive squags and Hall triple systems which were generalized by the author to the class of all finite nilpotent squags and their corresponding Steiner triple systems. In this article we present analogous theorems for nilpotent SQS-skeins and Steiner quadruple systems. We also generalize the wellknown doubling constructions of Doyen/Vandensavel and Armanious. It is then possible to describe the structure of all nilpotent Steiner quadruple systems completely: the nilpotent Steiner quadruple systems are exactly those obtained from the trivial 2-(or 4.) element Steiner quadruple system by repeated application of this generalized doubling construction. Moreover, we prove that the variety of semi-boolean SQS-skeins is not locally finite and contains non-nilpotent SQS-skeins.

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## Abstract A Steiner quadruple system of order __v__ (briefly an SQS(__v__)) is a pair (__X__,$\cal B$) with |__X__| = __v__ and $\cal B$ a set of quadruples taken from __X__ such that every triple in __X__ is in a unique quadruple in $\cal B$. Hanani [Canad J Math 12 (1960), 145–157] showed that