Semimodular Lattices and the Hall–dilworth Gluing Construction

We present a new gluing construction for semimodular lattices, related to the Hall–Dilworth construction. The gluing constructions in the lattice theory started with a paper of M. Hall and R. P. Dilworth [4] to prove that there exists a modular lattice that cannot be embedded in any complemented modular lattice. This construction is the following: let K and L be lattices, let F be a filter of K, and let I be an ideal of L such that F and I are isomorphic with φ : F → I. Then we form the disjoint union G = K ∪L and identify a ∈ F with aφ ∈ I, for all a ∈ F . a 5 b in G iff one of the following cases is satisfied: (i) a 5K b, a, b ∈ K, (ii) a 5L b, a, b ∈ L, (iii) a 5K z and zφ 5L b, a ∈ K, b ∈ L for some z ∈ F . I applied in my paper [5] the following special gluing construction (to give a very short proof for the theorem that every semimodular lattice of finite length has a cover-preserving embedding into a simple semimodular lattice), which is shown in Fig. 1. We define this gluing. Let L and K be semimodular lattices of finite length. Take a maximal chain C of L. Assume that K contains a filter C ′ isomorphic to C under the isomorphism ψ : C → C ′. Consider the attachment G of the lattice K to the lattice L over C by identifying C with C ′ along ψ, in the sense of G. Grätzer ∗This research was supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. K 77432.