The Lexicographic Product of Po-groups and n-Perfect Pseudo Effect Algebras

Abstract

We will study the existence of different types of the Riesz Decomposition Property for the lexicographic product of two partially ordered groups. A special attention will be paid to the lexicographic product of the group of the integers with an arbitrary po-group. Then we apply these results to the study of n-perfect pseudo effect algebras. We show that the category of strong n-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups with RDP 1 . 1

Kite Pseudo Effect Algebras

“…We note that an element a of a pseudo effect algebra E is infinitesimal if na = a + · · · + a (n-times addition of a) is defined in E for every integer n ≥ 1. We note that some families of perfect pseudo effect algebras were studied in [Dvu4,DvXi,DvKr].…”

Kite n-Perfect Pseudo Effect Algebras

“…By Proposition 3.3, the unital po-group (Z − → ⋉ φ G, u) satisfies RIP, too. Applying [DvKr,Thm 3.6], we have that the po-group (Z − → ⋉ φ G, u) satisfies RDP 1 as it was claimed. Finally, let us assume that G satisfies RDP 2 .…”

Lexicographic product vs $$\mathbb Q$$ Q -perfect and $$\mathbb H$$ H -perfect pseudo effect algebras

“…(ii) Let m 1 + m 2 > 0 and m 2 = 0. For the subcase n 2 > 0, we use the same ideas as case (i) of [DvKr,Prop 3.3,Thm 3.10], and we have the following decomposition table…”

“…Problem 1. Since both Q, R as well as any non-discrete H satisfy SRIP, according to [Go,Cor 2.12] or [DvKr,Prop 3…”