Incomplete idempotent Schröder quasigroups and related packing designs
type: Item
instance of: scholarly article
title: Incomplete idempotent Schröder quasigroups and related packing designs
zbMATH Open document ID: 0844.05027
DOI: 10.1007/BF01831143
author: Yanyan Li
published in: Aequationes Mathematicae
publication date: 1996-08-18T00:00:00Z
full work available at URL: https://eudml.org/doc/137633
review text: A Schröder quasigroup is a quasigroup satisfying the identity \((x\cdot y)\cdot (y\cdot x)= x\). In this paper the authors provide the 7 missing cases from the proof by \textit{C. J. Colbourn} and \textit{D. R. Stinson} [Edge-coloured designs with block size four, Aequationes Math. 36, No. 2-3, 230--245 (1988; Zbl 0658.05016)] to prove that there exists an idempotent Schröder quasigroup of order \(n\) if and only if \(n\) is a positive integer congruent to \(0\) or \(1\) modulo \(4\) and \(n\neq 5\) or \(9\). For \(n\equiv 2, 3\pmod 4\) no such quasigroup can exist, however the next closest thing is a incomplete idempotent Schröder quasigroup of order \(n\) with a hole of side 2 (an \(\text{IISQ}(n, 2)\)). The authors prove that there exists an \(\text{IISQ}(n, 2)\) for any positive integers \(n\equiv 2, 3\pmod 4\), \(n\geq 7\), except for \(n= 10\). The methods used in this paper are standard recursive and direct constructions.
reviewed by: J. H. Dinitz
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B40
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 856828
zbMATH Keywords: packing designs
zbMATH Keywords: Schröder quasigroup
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 856828