Lattice geometries
The Frink embedding theorem [\textit{O. Frink jun.}, Trans. Am. Math. Soc. 60, 452-467 (1946; Zbl 0060.058)] by which complemented modular lattices are embedded into a union of subspace lattices of projective planes and suitable irreducible projective coordinate spaces is generalized by Theorem 2 of \textit{U. Faigle} [Arch. Math. 36, 179-182 (1981; Zbl 0452.06005)] which says that any modular lattice can be embedded in a complete modular point lattice whose points (= join-irreducible elements) are compact and which satisfies all the identities of the embedded lattice. The authors give a generalization of Faigle's result: They define a \(J\)- additive lattice geometry to be a pair \((L,J)\), where \(L\) is a lattice and \(J\subset L\) such that any element in \(L\) is the join of all elements in \(J\) beneath it, and any \((x_ 1,x_ 2)\in L\times L\) is \(J\)-additive which means that whenever \(p\in J\), \(p\leq x_ 1\lor x_ 2\), there exist \(q_ 1,q_ 2\in J\), \(q_ i\leq x_ i\), \(i\in \{1,2\}\) such that \(p\leq q_ 1\lor q_ 2\). As examples of \(J\)-additive lattice geometries there are given among others (1) the lattice of submodules of any unital left module with \(J\) to be taken as the set of cyclic submodules, and (2) the biatomic lattices defined by the second author [Algebra Univers. 24, 60-73 (1987; Zbl 0643.06003)], where \(J\) is the set of atoms. The lattice of closed subspaces of an infinite-dimensional Hilbert space with \(J\) chosen to be the one-dimensional subspaces is not \(J\)-additive. Their main result is then: For any \(J\)-additive lattice geometry \((L,J)\) the underlying lattice \(L\) can be embedded under preservation of lattice identities into its lattice \(L'\) of flats which is then complete and algebraic. Furthermore \((L',J')\) is \(J'\)-additive, where \(J'\) denotes the image of \(J\) under the embedding. Here a flat \(F\) is defined to be subset of \(J\) such that \(x,y\in F\), \(z\in J\), \(z\leq x\lor y\) always implies \(z\in F\). Further results refer for instance to a lattice analogue of the Pasch axiom which is satisfied by \(J\)-additive lattice geometries or to exchange and antiexchange elements, cf. M. K. Bennett [loc. cit.].
scientific article; zbMATH DE number 488261
scientific article