On \(K\)-loops of finite order
type: Item
instance of: scholarly article
title: On \(K\)-loops of finite order
zbMATH Open document ID: 0803.20052
DOI: 10.1007/BF03323144
author: Alexander Kreuzer
author: Heinrich Wefelscheid
published in: Results in Mathematics
publication date: 1995-01-05T00:00:00Z
review text: This paper develops firstly the basic theory of \(K\)-loops and secondly there are given new construction methods for \(K\)-loops. For the axiomatic investigations the authors start from the quite general concept of a right loop \((L,+)\) (i.e., \(\forall a,b \in L\;\exists_ 1 x \in L : a+x=b\) and \(\exists 0 \in L : \forall a \in L : 0+a=a+0=a\)). Then for \(a,b \in L\) the maps \(a^ + : L \to L\); \(x \to a+x\) and \(\delta_{a,b} := ((a+b)^ +)^{-1} \circ a^ + \circ b^ +\) are permutations. The following properties (which allow to define the various types of loops, like Bol, Bruck or \(K\)-loops) are considered: \textbf{(K1\(\ell\))} \(\forall a,b \in L\;\exists_ 1 y \in L : y+a=b\), \textbf{(I)} \(a+b=0 \Rightarrow b+a=0\), \textbf{(K3)} \(\delta_{a,b} \in \text{Aut}(L,+)\), \textbf{(K4)} \(a+b=0 \Rightarrow \delta_{a,b}=\text{id}\), \(\text{\textbf{(K4)}}'\) \(\delta_{a,a}=\text{id}\), \textbf{(K5)} \((-a)+(-b)=-(a+b)\), \textbf{(K6)} \(\delta_{a,b}=\delta_{a,b+a}\), \textbf{(KB)} (Bol-identity) \(a+(b+(a+c))=(a+(b+a))+c\). Besides studying the connections between these properties and the consequences by assuming some of them, the authors stress their attention on right loops with \textbf{(K3)}. Then they add step by step further axioms and by assuming furthermore \textbf{(K1\(\mathbf\ell\))}, \textbf{(K5)} and \textbf{(K6)} they obtain the \(K\)-loops. To each right loop \((L,+)\) with \textbf{(K3)} and \textbf{(K4)} there correspond two groups, the ``structure group'' \(D := \langle \delta_{a,b} \mid a,b \in L\rangle\) of \((L,+)\) and according to \textit{G. Kist} [Result. Math. 12, 325-347 (1987; Zbl 0636.51012)] the group \(G := L \times D\) with \((a,\alpha) \cdot (b,\beta) := (a+\alpha(b),\delta_{a,\alpha(b)} \circ \alpha \circ \beta)\), the so-called quasidirect product of \(L\) and \(D\). Therefore also the reverse problem is studied, how one can obtain loops by starting from groups (\S 3). This leads the authors to new construction methods (\S 4) for l…
reviewed by: Helmut Karzel
Mathematics Subject Classification ID: 20N05
zbMATH DE Number: 554051
zbMATH Keywords: construction methods for \(K\)-loops
zbMATH Keywords: right loops
zbMATH Keywords: permutations
zbMATH Keywords: axioms
zbMATH Keywords: quasidirect products
zbMATH Keywords: finite proper loops
Wikidata QID: Q59383318
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 554051