# Boolean dimension and local dimension

## TL;DR

This paper investigates the relationships between boolean dimension, local dimension, and standard dimension in partially ordered sets, providing precise conditions under which these measures are bounded, thereby advancing understanding of poset complexity.

## Contribution

It precisely characterizes when bounds on one dimension measure imply bounds on the others, clarifying the interplay between these complexity parameters.

## Key findings

- Identifies conditions linking bounded boolean and local dimensions to standard dimension.
- Provides exact thresholds for when these dimensions are simultaneously bounded.
- Enhances understanding of poset complexity measures and their interrelations.

## Abstract

Dimension is a standard and well-studied measure of complexity of posets. Recent research has provided many new upper bounds on the dimension for various structurally restricted classes of posets. Bounded dimension gives a succinct representation of the poset, admitting constant response time for queries of the form "is $x<y$?". This application motivates looking for stronger notions of dimension, possibly leading to succinct representations for more general classes of posets. We focus on two: boolean dimension, introduced in the 1980s and revisited in recent research, and local dimension, a very new one. We determine precisely which values of dimension/boolean dimension/local dimension imply that the two other parameters are bounded.