Free Vector Lattices

Abstract

This note presents a useful explicit characterization of the free vector lattice FVL( ℵ ) on ℵ generators as a vector lattice of piecewise linear, continuous functions on Rℵ, where ℵ is any cardinal and R is the set of real numbers. A transfinite construction of FVL( ℵ ) has been given by Weinberg ( 14 ) and simplified by Holland ( 13 , § 5). Weinberg's construction yields the fact that FVL( ℵ ) is semi-simple; the present characterization is obtained by combining this fact with a theorem from universal algebra due to Garrett Birkhoff.