Ternary fields of compact projective planes
type: Item
instance of: scholarly article
title: Ternary fields of compact projective planes
zbMATH Open document ID: 0652.51017
DOI: 10.1007/BF02941602
author: Theo Grundhöfer
published in: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
publication date: 1987-01-01T00:00:00Z
review text: This paper gives a characterization of those ternary fields which belong to compact topological projective planes. Let \((K,\tau)\) be a ternary field, \(\tau\) its ternary operation, \({\mathcal O}\) a locally compact topology on \(K\), and \({\mathfrak P}:=(P,{\mathcal L})\) its projective coordinate plane. The author carefully analyzes the problem of ``extending'' \({\mathcal O}\) from \(K\) to \(P\) and \({\mathcal L}\). He shows (1.1,1.2) that \(P\) (and dually \({\mathcal L}\)) can be given a topology \({\mathcal O}_ P\), (resp. \({\mathcal O}_{{\mathcal L}}\)) such that some three affine point sets which cover \(P\) are openly embedded if \((K,\tau,{\mathcal O})\) satisfies the following condition: (W) \(xy:=\tau (x,y,0)\) and \(\dot b(x,y,0)\) depend continuously on \(x,y\in \bar K:=K\cup \{\infty \}\) (one-point-compactification of \(K\)), \((\infty,0)\neq (x,y)\neq (0,\infty)\), where \(\dot b(x,m,y):=b \Leftrightarrow y=\tau (x,m,b)\). Furthermore (1.3), he describes convergence of nets in \(P\) and \({\mathcal L}\) by convergent nets in \(\bar K,\) cf. also \textit{S. Arumugam} [Kyungpook Math. J. 26, 217--222 (1986; Zbl 0622.51009)]. Then the following conditions on \((K,\tau,{\mathcal O})\) are considered: (A) \(\tau: K^ 3\to K\) and \(\dot b: K^ 3\to K\) are continuous. (B) For all nets \(x_{\nu}\to \infty\), \(m_{\nu}\to m\in K\), \(b_{\nu}\to b\in K:\) \(x_{\nu}\setminus \tau (x_{\nu},m_{\nu},b_{\nu}]\to m\) (\(x\setminus y\) is defined by \(x(x\setminus y)=y).\) (C) For all nets \(m_{\nu}\to \infty\), \(x_{\nu}\to x\in K\), \(y_{\nu}\to y\in K:\) \(\dot x(m_{\nu},\dot b(x_{\nu},m_{\nu},y_{\nu}))\to x\) \((\dot x(m,b):=x \Leftrightarrow 0=\tau (x,m,b).\) (D) For all nets \(x_{\nu}\to \infty\), \(m_{\nu}\to \infty\), \(t_{\nu}\to t\in K:\) \(x_{\nu}\setminus \tau (x_{\nu},m_{\nu},\dot b(t_{\nu},m_{\nu},0))\to \infty.\) Sharpening the assumptions on \((K,\tau,{\mathcal O})\) yields the desired characterization (2.2): If \((K,\tau,{\mathcal O})\) satis…
Mathematics Subject Classification ID: 51H05
Mathematics Subject Classification ID: 12J99
Mathematics Subject Classification ID: 12K99
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 17D99
zbMATH DE Number: 4063922
zbMATH Keywords: locally compact ternary field
zbMATH Keywords: locally compact quasifield
zbMATH Keywords: compact topological projective planes
reviewed by: Horst Szambien
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4063922