A  Riesz space E can be defined to be a vector space endowed with a partial order  ≤, that for any x, y, z in E, satisfies:

1.  x ≤ y implies x + z ≤ y + z.
2.  For any scalar 0 ≤ α, x ≤ y implies αx ≤ αy.
3. For any pair of vectors x, y in E there exists a supremum (denoted x ∨ y) in E with respect to the partial order (≤).

Riesz spaces are named after F Riesz who first defined them in his 1928 paper  Sur la décomposition des opérations fonctionelles linéaires, Atti congress. internaz. mathematici (Bologna, 1928), 3, Zanichelli (1930) pp. 143–148ires.  Riesz spaces are also called  lattice ordered vector spaces or vector lattices.  Riesz spaces have wide applications specially in economics and preference modeling.