# On the linking of number lattices

## TL;DR

This paper explores the structure and properties of number lattices using topological network ideas, introducing linking operations, duality theorems, and algorithms for constructing and analyzing lattices in a mathematical framework.

## Contribution

It introduces a linking operation for generalized number lattices, proves duality and inversion theorems, and develops algorithms for constructing and analyzing lattices with applications in network theory.

## Key findings

- Linked number lattices can be combined and inverted using the proposed operations.
- Duality theorems enable construction of self-dual lattices from existing ones.
- Algorithms are provided for basis reduction in lattice constructions involving unimodular matrices.

## Abstract

In this paper we study ideas which have proved useful in topological network theory in the context of lattices of numbers. A number lattice $L_S$ is a collection of row vectors, over $\mathbb{Q}$ on a finite column set $S,$ generated by integral linear combination of a finite set of row vectors. A generalized number lattice $K_S$ is the sum of a number lattice $L_S$ and a vector space $V_S$ which has only the zero vector in common with it. The dual $K^d_S$ of a generalized number lattice is the collection of all vectors whose dot product with vectors in $K_S$ are integral and is another generalized number lattice. We consider a linking operation ('matched composition`) between generalized number lattices $K_{SP},K_{P}$ (regarded as collections of row vectors on column sets $S\cup P, P,$ respectively with $S,P,$ disjoint) defined by $K_{SP}\leftrightarrow K_{P}\equiv \{f_S:((f_S,g_P)\in K_{SP}, g_P \in K_{P}\}.$ We show that this operation together with contraction and restriction, and the results, the implicit inversion theorem (which gives simple conditions for the equality $K_{SP}\leftrightarrow (K_{SP}\leftrightarrow K_S)= K_S,$ to hold) and implicit duality theorem ($(K_{SP}\leftrightarrow K_{P})^d= K_{SP}^d\leftrightarrow K_{P}^d$)), are both relevant and useful in suggesting problems concerning number lattices and their solutions. Using the implicit duality theorem, we give simple methods of constructing new self dual lattices from old. We also give new and efficient algorithms for the following. Given $V_{SP},K_P,$ such that $V_{SP}\leftrightarrow (V_{SP}\leftrightarrow K_P)= K_P,$ where $V_{SP}$ is a vector space with a totally unimodular basis matrix, to construct reduced bases for the number lattice part of $V_{SP}\leftrightarrow K_P, K_P^d, (V_{SP}\leftrightarrow K_P)^d,$ from a reduced basis for the number lattice part of $K_P.$