Ternary fields of compact projective planes
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 \((K,\tau,{\mathcal O})\) satisfies (W), (A), (B), (C), (D), then \(({\mathfrak P},{\mathcal O}_ P,{\mathcal O}_{{\mathcal L}})\) is a compact topological projective plane and any such plane arises this way. (3.1): For quasifields \((Q,+,\cdot)\) its enough to assume (W'), (A'), (B'), where (W') For all nets \(x_{\nu}\to x\in Q^*\), \(y_{\nu}\to \infty\), one has \(x_{\nu}y_{\nu}\to \infty\); (A') Addition and multiplication are continuous on \(Q^ 2\); (B') For all nets \(x_{\nu}\to \infty\), \(m_{\nu}\to m\in Q\), \(b_{\nu}\to b\in Q\), one has \(x_{\nu}\setminus (x_{\nu}m_{\nu}+b_{\nu})\to m\). If \(Q\) is even non-discrete and finite-dimensional over some subfield of its kernel, then (W') and the continuity of multiplication on \(Q^2\) suffice, (3.2): cf. \textit{H. Hähl}, Arch. Math. 38, 273--279 (1982; Zbl 0485.22005). Since any non-discrete locally compact topological planar nearfield satisfies (W') and (B'), such nearfields also yield compact topological projective planes, (3.3). It is an open problem whether every locally compact topological ternary field \((K,\tau,{\mathcal O})\) in the sense of \textit{H. Salzmann} [Adv. Math. 2, 1--60 (1967; Zbl 0153.21601); 7.5] gives rise to a compact topological projective plane. The author characterizes such ternary fields by \((D^+)\) There are no nets \(x_{\nu}\to \infty\), \(m_{\nu}\to \infty\), \(a_{\nu}\to a\), \(y_{\nu}\to y\), \(c_{\nu}\to c\), \(d_{\nu}\to d\) with \(a,y,c,d\in K\) such that \(\tau (x_{\nu},m_{\nu},\dot b(a_{\nu},m_{\nu},y_{\nu}))=\tau (x_{\nu},c_{\nu},d_{\nu})\) for all \(\nu\). In two importance cases \((D^+)\) is shown to hold: (i) \((K,{\mathcal O})\) is connected, (4.2); this reproves the famous result of L. A. Skornyakov and H. Salzmann, see \textit{H. Salzmann} [loc. cit. 7.14, 7.15]. (ii) \((K,\tau,{\mathcal O})\) coordinatizes a topological affine plane plane having compact line pencils, (6.1); this improves theorem 7 of \textit{L. A. Skornyakov} [Tr. Mosk. Mat. Ob.-va 3, 347--373 (1954; Zbl 0057.36201)].
scientific article; zbMATH DE number 4063922
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