# Lexicographic cones and the ordered projective tensor product

## TL;DR

This paper introduces lexicographic cones to generalize ordered vector spaces and uses them to establish properties of tensor cones and characterize finite-dimensional vector lattices.

## Contribution

It develops lexicographic cones for posets, proving the projective tensor cone is a cone and providing a new characterization of finite-dimensional vector lattices.

## Key findings

- The projective tensor cone of two cones is itself a cone.
- Lexicographic cones generalize standard lexicographic cones.
- Finite-dimensional vector lattices are characterized in a new way.

## Abstract

We introduce lexicographic cones, a method of assigning an ordered vector space $\Lex(S)$ to a poset $S$, generalising the standard lexicographic cone. These lexicographic cones are then used to prove that the projective tensor cone of two arbitrary cones is a cone, and to find a new characterisation of finite-dimensional vector lattices.