E. Fried

Projective geometries as cover-preserving sublattices

It is a well known result in the folklore of lattice theory that whenever M 3 (the five element modular nondistributive lattice) can be embedded into a finite modular lattice L, then M 3 also has a cover-preserving embedding into L, that is, an embedding ep with the property that if a covers b in M 3 , then ep(a) covers ep(b) in L. Therefore, if L is a finite nondistributive modular lattice, then L contains M3 as a cover-preserving sublattice. We formalize this concept: a finite lattice K has the cover-preserving embedding property, abbreviated as CPEP, with respect to a variety V of lattices, if whenever K can be embedded into a finite lattice L in V, then K has a cover-preserving embedding into L. In this note, we determine which finite projective geometries P satisfy the CPEP with respect to the variety M of modular lattices; from our point of view, a finite projective geometry is a finite complemented simple modular lattice. Parts of our main result are in the folklore; certainly, specialists in the field knew that M 3 has CPEP while M n , n > 3, does not. Those familiar with the deep results of A. Huhn on diamonds and of R. Freese on n-frames could easily deduce our results for projective geometries of dimension at least three. It appears to us that those techniques do not apply to projective planes. General references on projective geometries are [1, Chapter 13] and [6, Section IV.5]. We make particular use of two facts: a projective geometry of length ~ 4 (i.e., of geometric dimension ~ 3) is arguesian; any finite arguesian projective geometry of length m ~ 3 is isomorphic to the lattice Sub (V) of all subspaces of an

Pasting and modular lattices

A classical lattice construction of R. P. Dilworth is the gluing of two lattices. A number of recent papers by A. Slavik, A. Day, and J. Jezek investigated a generalization: pasting. In this note we prove that by pasting two finite modular lattices, one obtains a modular lattice. A classical problem of lattice theory (see, e.g., [3, Problems V. 10 and V. 1]) has recently been solved: Only three lattice varieties have the Amalgamation Property, namely, T, the trivial variety, D, the variety of distributive lattices, and L, the variety of all lattices. This result was proved in two steps. First, G. Gratzer, B. J6nsson, and H. Lakser [4] proved that T and D are the only modular lattice varieties that have the Amalgamation Property. Then A. Day and J. Jezek [1] proved that L is the only nonmodular lattice variety with the Amalgamation Property. The method of G. Gratzer, B. J6nsson, and H. Lakser [4] is projective geometric: in a modular lattice variety V having the Amalgamation Property, a lattice L is amalgamated many times with M3, the five element modular, nondistributive lattice, obtaining a complemented modular lattice. Then Frinks Theorem [2] gives an embedding of L into a projective geometry. The method of A. Day and J. Jezek [1] uses a version of amalgamation discussed in A. Slavik [8] (see also G. Gratzer [4, Exercise 12 of Section V.4]); in this paper it will be called pasting (see below for the definition). Pasting is stronger than gluing but much weaker then amalgamation. They prove the following theorem: Let V be a nonmodular variety of lattices; if V is closed under the pasting of finite lattices, then V = L. A. Day and J. Jezek raise the following problem: Can one prove the result of G. Gratzer, B. J6nsson, and H. Lakser on the nonexistence of modular nondistributive lattice varieties having the amalgamation property using this method. More specifically: can one start with M3, and obtain N5 (the five element Received by the editors October 30, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 06B05, 06C05; Secondary 08B25.

Multipasting of lattices

In this paper we introduce a lattice construction, calledmultipasting, which is a common generalization of gluing, pasting, andS-glued sums. We give a Characterization Theorem which generalizes results for earlier constructions. Multipasting is too general to prove the analogues of many known results. Therefore, we investigate in some detail three special cases: strong multipasting, multipasting of convex sublattices, and multipasting with the Interpolation Property.