Global-affine morphisms of projective lattice geometries
Generalizations of the fundamental theorem of projective geometry considered in the framework of lattice theory have been given for instance by \textit{V. P. Camillo} [Rocky Mt. J. Math. 14, 475-486 (1984; Zbl 0543.16023)], and they are the aim of the paper under review. The author presents results to be proven in his theories which rely on the notion of projective lattice geometry [see the author and \textit{S. E. Schmidt}, Geom. Dedic. 43, No. 3, 243-264 (1992; Zbl 0787.51004)]. These geometries are triples \(G= (L,E,F)\) where \(L\) is a complete lattice and \(F\), \(E\) are sets of compact elements in \(L\) such that \(F\subseteq E\) and some further conditions are satisfied which are general enough to include the lattice \(L\) of submodules of some unitary left module \(M\) over some ring \(R\) as an example. In this case \(E\) is chosen to be the set of those submodules which are isomorphic to \(R\) considered as a module over itself. Geometries \(G\) represented this way are called module-induced. A mapping \(\alpha: L\to L'\) where \(G= (L,E,F)\), and \(G'= (L',E',F')\) are projective lattice geometries is called global-affine homomorphism if it preserves arbitrary joins, finite meets and pairs \((f,h)\) of complemented elements from \(L\) with \(f\in F\). Now (Theorem 3.1), global-affine morphisms \(\alpha\) between geometries induced by free modules of rank \(\geq 3\) are shown to be induced by some basis preserving semilinear mapping between the underlying modules at least when \(\alpha\) is restricted to the set of complements of some hyperplane. These restrictions do not occur if \(\alpha\) is an isomorphism, and additional conditions on the modules are assumed, e.g. if (4.3) the modules are free of infinite rank or (4.5) are free over a proper right Bézout ring of stable rank 2.
scientific article; zbMATH DE number 446735
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