A geometric description of modular lattices
The authors consider modular lattices \(M\) which are spatial, i.e. every element is a join of some subset of the set \(S(M)\) of points (= completely join irreducible elements) of \(M\). If \(S(M)\) is given the induced partial order \(\leq\) and a collinearity relation \(C\) where \(C(p, q,r)\) is defined by \(p+ q= p+r= q+r\) for any pairwise incomparable \(p,q,r\in S(M)\) then \(S(M)\) constitutes a so-called ordered space in which subspaces are then defined as those order ideals which are closed with respect to collineraity. The resulting lattice of subspaces \(L(S(M))\) contains \(M\), and it is even isomorphic to \(M\) if \(M\) is algebraic (2.4). Taking up investigations from \textit{U. Faigle} and the first author [Trans. Am. Math. Soc. 266, 319-332 (1981; Zbl 0466.51001)] an ordered space \((S,\leq C)\) is called projective if it satisfies a line regularity axiom and a kind of a Veblen-Young axiom which includes some degenerate situations. Such ordered spaces are shown to arise exactly from spatial modular lattices (3.2, 3.3). Furthermore, any variety of modular lattices is generated by subspace lattices of countable ordered projective spaces (6.2). The work is done in continuation of the first author and \textit{M. Wild} [J. Algebra 136, 17-36 (1991; Zbl 0725.16014)] by carefully selecting necessary conditions (chain conditions, countability) on the point set and so-called line bases which allow to reconstruct the lattice structure in the case of modular algebraic spatial lattices. Finally, the authors point out properties which are equivalent to 2- distributivity in modular spatial lattices \(M\); one of these properties is the absence of the triangle configurations occurring in the Veblen- Young axiom (7.2), see also \textit{B. Jónsson} and \textit{J. B. Nation} [Acta Sci. Math. 51, 123-128 (1987; Zbl 0633.06002)]. They show that as in the case of finite length, treated by B. Jonsson and J. B. Nation [loc. cit], 2-distributive lattices can be embedded in subspace lattices of vector spaces over fields of arbitrary prime characteristic (1.1). They infer (1.2) that for different primes the varieties generated by such subspace lattices intersect only in the class of 2-distributive modular lattices.
scientific article; zbMATH DE number 590886
scientific article