The Causal Poset is Directed but not Lattice Ordered

TL;DR

This paper investigates the structure of the causal set poset in discrete quantum gravity, proving it is directed but not lattice ordered, which impacts understanding of universe growth models.

Contribution

It demonstrates that the causal set poset is directed but provides a counterexample showing it is not lattice ordered, clarifying its mathematical properties.

Findings

The causal set poset is directed.

The causal set poset is not lattice ordered.

Counterexample illustrating non-lattice structure.

Abstract

In the causal set approach to discrete quantum gravity the universe grows one element at a time in discrete steps. At each step the process has the form of a causal set (causet) and the "completed" universe is given by a path through a discretely growing chain of causets. The collection of causets forms a partially ordered set (poset) in a natural way. We first show that this poset is directed. We then give a counterexample which shows it is not lattice ordered.