Lexicographic cones and the ordered projective tensor product

We introduce lexicographic cones, a method of assigning an ordered vector space $\text{Lex}(S)$ 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.