Revisiting the free 2-generator abelian ℓ-group

The free n-generator abelian ℓ-group (Fn,P) was characterized by G. Birkhoff as the smallest ℓ-group of real-valued functions over image containing the n identity functions, and closed under the pointwise operations of addition, subtraction, max and min. It follows that Fn is a subdirect product of integers. Since Fn is countable, by a theorem of Baer and Specker, Fn is a free abelian group, image for some generators χi. There is no direct description in the literature of those elements n1χ1+cdots, three dots, centered+nkχk which belong to the monoid P of positive elements of Fn. A simple description is given in this elementary note for the case n=2.