Definition 1

An ordered vector space is said to be a Riesz space (or lattice-ordered vector space)$^1$ if its order structure is a lattice-ordering (that is, if every pair of elements $x, y$ of E has a supremum $\sup(x, y)$ and an infimum $\inf(x, y)$ ).$^2$

(1) Also called a "vector lattice".

(2) Borne supérieure (supremum) is also translated as "least upper bound" (S, R, §6, 7). Similarly borne inférieure (infimum) is also translated as "greatest lower bound".