This note gives answer to problem 103 in G. Birkhoff's "Lattice Theory" (revised edition, 1948). The problem is:Let G be the additive group of all real vectors (a, b), ordered by defining (a, b)≧(c, d) to mean "either a>c, or a=c and b≧d." Is it possible to define multiplication in G so as to make it into an ordered ring?The answer is positive. Our results and discussions are summarized below.In §1, it is pointed out that G can be made into an ordered ring by each of the following types of multiplication: (a, b) (c, d) = (0, acβ), β≧0.(Ⅰ) (a, b) (c, d) = (acα, g(ac)), α>0.(Ⅱ) (a, b) (c, d) = (acα,bcα),α>0.(Ⅲ) (a, b) (c, d) = (acα, adα), α>0.(Ⅳ)(a, b) (c, d)= (acα, acβ+adα+bcα),α>0,β arbitrary.(Ⅴ) where α, β are arbitrary constants within restrictions given in each case, and g(x) in (Ⅱ) is any one of a class of functions defined as follows:Assuming the axiom of choice, and hence assuming that the set of real numbers can be arranged into a Well-ordered set W. Define g(x) on W by transfinite induction: Suppose that g(x) has been consistently defined for all numbers preceding aλ in W. (D_1) If aλ can be "rationally expressed" by a finite set of numbers preceding it in W, i.e. if aλ=r_1 aλ_1 +…+ r_k aλ_k(r_1, ..., r_k rational; k finite) for some aλ_1, …, aλ_k preceding aλ in W, then take any one of such expressions, say the above one, and define g(aλ) =r_1 g(aλ_1)+... + r_kg(aλ_k) . (D_2) If no such expression exists, then take a real number β_λ arbitrarily and define g(aλ) = aλβ_λ.The consistency of the above definition is proved. And it is pointed out that what we have defined is the widest class of real functions satisfying the condition g(a+b)=g(a)+g(b) . (including linear functions "ax" as special cases.)In §2, we consider the question: How many possible types of multiplication are there on G?Suppose that G has been made into an ordered ring, in which (1,0)~2= (α,β), (0,1)~2= (Υ, δ), (1,0) (0,1)=(ζ, η), (0,1) (1,0)=(θ,κ). It is shown that Υ=δ=ζ=θ=0, αη=η~2, αk=k~2, α≧0,and β≧0 when α=0 . Then the discussion proceeds along several cases:When α=η=k=0, β≧0, the multiplication of any two vectors in G satisfies the formula: (α,b) (c,d)=(0, αcβ). (Ⅰ)When α>0,η=κ=0, the multiplication of any two vectors in G satisfies the formula: (a, b) (c,d)=(acα, g(ac)); (Ⅱ) where g(x) is any one of the functions defined in §1 and satisfying g(1)=β.In other cases, only partial results are obtained, the complete determination of the multiplication formulas seems difficult.In §3, we examine the isomorphism or non-isomorphism between the ordered rings having G as additive group. It is found that a large part (nearly all) of them fall into six classes of isomorphic ordered rings. (The automorphism groups of five of these classes are also determined.) As a consequence of these considerations, some of the unsatisfactory points (e.g. the use of the axiom of choice in defining (Ⅱ).) in the above sections become algebraically nonessential.
