A class of fibered loops
type: Item
instance of: scholarly article
title: A class of fibered loops
zbMATH Open document ID: 0643.20048
DOI: 10.1007/BF03322411
author: Elena Zizioli
published in: Results in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: A fibered loop (P,\({\mathcal F},\cdot)\) is a loop (P,\(\cdot)\) together with a set \({\mathcal F}\) of its subloops pairwise intersecting only in the identity element of (P,\(\cdot)\) and such that \(\cup {\mathcal F}=P\). The aim of this paper is to determine fibered loops which are not groups. The author presents a method to construct such structures for any order \(n\geq 4\) and for countable order. The construction stems from a class of semi-diagonal latin squares \([a_{ij}]\) of order m (i.e. such that \(a_{ii}=i\) for any \(i\in \{1,2,...,m\})\) whose matrices are explicitly determined for any order m. Some properties of the so obtained fibered loops are investigated, in particular it turns out that this method gives proper fibered loops for any order \(n>4\) and that such fibered loops (P,\({\mathcal F},\cdot)\) carry a geometric structure of incidence space if and only if every element of \({\mathcal F}\) is a subgroup of order 2.
reviewed by: Elena Zizioli
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 51E30
Mathematics Subject Classification ID: 05B15
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 4047972
zbMATH Keywords: fibered loops
zbMATH Keywords: semi-diagonal latin squares
zbMATH Keywords: incidence space
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf03322411
OpenAlex ID: W2085074454
description: scientific article; zbMATH DE number 4047972