Revisiting the free 2-generator abelian ℓℓ-group

The free nn-generator abelian ℓℓ-group (Fn,P)(Fn,P) was characterized by G. Birkhoff as the smallest ℓℓ-group of real-valued functions over RnRn containing the nn identity functions, and closed under the pointwise operations of addition, subtraction, max and min. It follows that FnFn is a subdirect product of integers. Since FnFn is countable, by a theorem of Baer and Specker, FnFn is a free abelian group, Fn=⨁i∈N〈χi〉Fn=⨁i∈N〈χi〉 for some generators χiχi. There is no direct description in the literature of those elements n1χ1+⋯+nkχkn1χ1+⋯+nkχk which belong to the monoid PP of positive elements of FnFn. A simple description is given in this elementary note for the case n=2n=2.