Riesz ideals in generalized pseudo effect algebras and in their unitizations

• DOI: 10.1007/s00500-009-0412-6
• Publication Date: 2009-03-18T03:25:16Z
• Authors: Yongjian Xie; Yongming Li
• Keywords: Riesz representation theorem; Congruence relation; Isomorphism (crystallography); Riesz potential; Riesz transform; Lattice (music)

We prove that there is an order isomorphism between the lattice of all normal Riesz ideals and the lattice of all Riesz congruences in upwards directed generalized pseudoeffect algebras (or GPEAs, for short). We give a sufficient and necessary condition under which a normal Riesz ideal I of a weak commutative generalized pseudoeffect algebra P is a normal Riesz ideal also in the unitization $$\widehat{P}$$of P. These results extend those obtained recently by Avalllone, Vitolo, Pulmannova and Vincekova for effect algebras. At the same time, we give the conditions under which the quotient of a generalized pseudoeffect algebra P is a generalized effect algebra and linearly ordered generalized pseudoeffect algebra.