S-Glued sums of lattices

TL;DR

This paper introduces the S-glued construction, a method to build complex lattices from a family of smaller lattices, enabling new representations of finite-length modular lattices via projective geometries.

Contribution

It generalizes the Hall-Dilworth lattice construction to a broader family of lattices indexed by a set S, allowing for more flexible and complex lattice structures.

Findings

The S-glued method can construct lattices with specified local and global properties.

It provides a new way to represent finite-length modular lattices.

Application to projective geometries demonstrates its utility.

Abstract

For many equation-theoretical questions about modular lattices, Hall and Dilworth give a useful construction: Let $L_0$ be a lattice with largest element $u_0$, $L_1$ be a lattice disjoint from $L_0$ with smallest element $v_1$, and $a \in L_0$, $b \in L_1$ such that the intervals $[a, u_0]$ and $[v_1, b]$ are isomorphic. Then, after identifying those intervals you obtain $L_0 \cup L_1$, a lattice structure whose partial order is the transitive relation generated by the partial orders of $L_0$ and $L_1$. It is modular if $L_0$ and $L_1$ 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 $L_x\ (x \in S)$ of lattices can specify a lattice with the small-scale lattice structure determined by the $L_x$ and the large-scale structure determined by $S$. A crucial application is representing finite-length modular lattices using projective geometries.