2024/09/16 by Christian Herrmann, Herrmann, Christian, Dale R. Worley +1
Computer Science · #06B15 #Advanced Algebra and Logic #Combinatorics (math.CO) #FOS: Mathematics #Rough Sets and Fuzzy Logic
paper · pdf · doi:10.48550/arxiv.2409.10738
openalex publication_date 2024/09/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let L0 be a lattice with largest element u0, L1 be a lattice disjoint from L0 with smallest element v1, and a ∈ L0, b ∈ L1 such that the intervals [a, u0] and [v1, b] are isomorphic. Then, after identifying those intervals you obtain L0 ∪ L1, a lattice structure whose partial order is the transitive relation generated by the partial orders of L0 and L1. It is modular if L0 and L1 are modular. Since in this construction the index set \0, 1\ is essentially a chain, this work presents a method -- termed S-glued -- whereby a general family Lx (x ∈ S) of lattices can specify a lattice with the small-scale lattice structure determined by the Lx and the large-scale structure determined by S. A crucial application is representing finite-length modular lattices using projective geometries.