On some special classes of partially ordered linear algebras

Elementary Functional Equations on Partially Ordered Linear Algebra

“…Let I = {y E A: y > I and y~' > 0} and then define A, = I -I. It is shown in [5] that A, is an order-closed and order-convex commutative subalgebra of A ; furthermore, it is an /-ring. In fact, A, is algebraically and order isomorphic to an algebra of continuous almostfinite extended-real-valued functions defined on a compact Hausdorff space [9].…”

“…For detailed information on A,, see [5]; in particular, we refer to the interesting examples on pages 657, 679 and 680 in [5]. It should be noted that the term polac instead of dsc-pola was used in [5]. Lemma 1.1.…”

“…See Theorems 1.6.3 and 1.6.10 in [5]. □ In this paper, there are two frequently used subsets in A which we will denote by H = {« G A: u > 1}, K = {x G A: x > 0}.…”

Remarks on H. H. Schaefer’s conditions

“…Work of this type has been done by DeMarr [2], [3]. These two conditions are motivated by Schaefer's work [5, p. 169]; see also [4, p. 2108].…”

Representation of partially ordered linear algebras

“…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.…”