Representation of partially ordered linear algebras

Abstract

In [2] and [3] a condition on partially ordered linear algebras (pola's) is defined, and it is shown that Dedekind cr-complete polas satisfying this condition have many of the properties of function spaces. Using a theorem of H. Nakano we can show, even without the hypothesis that the pola is Dedekind cr-complete, that any such pola is isomorphic to a pola of continuous, almost-finite, extended-real-valued functions. If A is a pola with multiplicative identity 1 the condition mentioned is:Pi. If x e A and x^. 1, then x has an inverse and x _1^0 . PROOF. The standard completion procedure for Archimedean ordered linear spaces shows that A is isomorphic with an order dense subspace  of a Dedekind complete linear lattice D. In [4, p. 150] it is shown that the multiplication on  can be extended to D in such a way that D is a pola if the following continuity condition is satisfied: For every subset B of A, inf 5=0 implies inf(âL8)=inf(2ta)==0 for all positive elements a in A, Given P l5 multiplication by (a+1) _1 shows this condition is satisfied. Thus D is a linear lattice pola and the order density of  shows (since 1 is easily seen to be a weak order unit for A) that the image of 1 is a weak order unit for D. Now D (and hence A) has a representation of the type desired by [1, Corollary, p. 625]. THEOREM. In order for an Archimedean pola A with identity 1 to be isomorphic to a pola of continuous, almost-finite, extended-real-valued functions on a compact Hausdorff space X, it is sufficient thatTo prove the second statement we note that the assumptions, together with the Stone-Weierstrass theorem, give the result that if A-+ is the isomorphism then  1 =C(X). Then, given any x in A such that x_l, (1970). Primary 06A70; Secondary 06A65. AM S (MOS) subject classifications

Isotone functions on partially ordered linear algebras with a multiplicative diagonal map

“…Any dsc-pola A with a multiplicative DP map A always admits a function with the logarithm (root) property which we explain as follows. In [6] it was proved that A, is algebraically and order isomorphic to an algebra S of continuous, almost-finite, extended-real-valued functions defined on a compact Hausdorff space. Hence, we may identify A, with S. It is clear then for 1 < z E Ax, ln(z) and Vz also belong to Ax.…”