The variety of modular lattices is not generated by its finite members

Abstract

Abstract. This paper proves the result of the title. It shows that there is a five-variable lattice identity which holds in all finite modular lattices but not in all modular lattices. It is also shown that every free distributive lattice can be embedded into a free modular lattice. An example showing that modular lattice epimorphisms need not be onto is given.We prove the result of the title by constructing a simple modular lattice of length six not in the variety generated by all finite modular lattices. This lattice can be generated by five elements and thus the free modular lattice on five generators, FM (5), is not residually finite.Our lattice is constructed using the technique of Hall and Dilworth [9] and is closely related to their third example. Let F and K be countably infinite fields of characteristics p and q, where p and q are distinct primes. Let Lp be the lattice of subspaces of a four-dimensional vector space over F, Lq the lattice of subspaces of a four-dimensional vector space over K. Two-dimensional quotients (i.e. intervals) in both lattices are always isomorphic to Mu (the two-dimensional lattice with « atoms). Thus Lp and Lq may be glued together over a two-dimensional quotient via [9], and this is our lattice. Notice that if F and K were finite fields we could not carry out the above construction since two-dimensional quotients of Lp would have/)" + 1 atoms and those of Lq would have qm + 1, for some n, m > 1. However these numbers are never equal. To some extent the proof is based on this fact. We prove our result by letting / be a homomorphism from a modular lattice M onto our lattice L. A great deal of the structure of L can be pulled back through / into M. We then assume M is residually finite and using von Neumann's theorem arrive at a contradiction similar to the one described above. K. Baker [1] and R. Wille [20] have constructed varieties of modular lattices not generated by their finite members. Using a lattice constructed by E. T. Schmidt [18] the author has shown there is a variety of modular lattices not even generated by its finite dimensional members [6]. However, Schmidt's

The ring of an outer von Neumann frame in modular lattices

“…In order to mimic the classical coordinatization of projective geometries in modular lattice environment, he associated a ring, the so-called coordinate ring, with each n-frame. Although von Neumann assumed that L is a complemented modular lattice and n ≥ 4, his construction of the coordinate ring (without coordinatization) extends to arbitrary modular lattices without complementation, see Artmann [1] and Freese [6], and even to n = 3 if L is Arguesian, see Day and Pickering [4]. The equational theory of frame generated modular lattices is given by Herrmann [9].…”

“…Frames and Huhn diamonds are used in the proof of several deep results showing how complicated modular lattices are, we mention only Freese [6], Huhn [12], and Hutchinson [14]. Frames or Huhn diamonds were also used in the theory of congruence varieties, see [15], [2], and Freese, Herrmann, and Huhn [7].…”

Many varieties of lattices with nonsurjective epimorphisms

“…However, it is easy to produce nonsurjective epimorphisms in the variety D of distributive lattices. The variety M of modular lattices also has nonsurjective epimorphisms, as Freese showed in Theorem 3.3 of [5].…”

The product of two von Neumann n-frames, its characteristic, and modular fractal lattices

“…It belongs to the folklore of lattice theory that Sub(Z n p ) is a subdirectly irreducible lattice; cf., e.g., Freese [5] or Grätzer [11, page 240] or Herrmann [12]. Hence the famous lemma of Jónsson [17] yields that L has an ultrapower L ′ and L ′ has a sublattice K such that Sub(Z n p ) is a homomorphic image of K. Now we will use Theorem 1.6 of Freese [6]; note that we could use Freese [5] instead. Freese's theorem says that for any g > 1, n-frames of characteristic g are projective configurations in modular lattices.…”