Whom is the decomposition theorem for finite-dimensional Riesz spaces (vector lattices) is attributed to?

Question

There is a result in the theory of Riesz spaces suggesting that every such finite-dimensional space over R, is isomorphic to a direct sum of orthogonal ideals, each one being a lexicographically ordered vector lattice; and such decomposition is unique except for the sequence of the direct sum components.

I would like to know who this result belongs to. Or at least a specific attribution of a more generic result (to which the above mentioned one is corollary). Unfortunately, I cannot track it down.

Thank you for your help!

Answer 1

According to the Notes in Schaefer - Banach Lattices and Positive Operators, pg. 138,

The result that every finite dimensional vector lattice can be built up from $\mathbb{R}$ by "direct and lexicographical union" is found in Birkhoff [1967]. On the other hand, the introduction of the radical and its systematic use, appear to be new.

Birkhoff in his book Lattice Theory proves it without attributing it to someone, while generally speaking he tries to include references whenever he is able to. So it looks fair to attribute it to both Birkhoff and Schaefer.