Decomposition of Pseudo-effect Algebras and the Hammer–Sobczyk Theorem

Abstract

We prove an algebraic and a topological decomposition theorem for complete pseudo-D-lattices (i.e. lattice-ordered pseudo-effect algebras). As a consequence, we obtain a Hammer–Sobczyk type decomposition theorem for group-valued modular measures defined on pseudo-D-lattices and compactness of the range of every (Formula presented.)-valued σ-additive modular measure on a σ-complete pseudo-D-lattice. © 2015 Springer Science+Business Media Dordrech

Pseudo-D-lattices and separating points of measures

“…In this case h(1) is also called the height of L. Note that an element a in a modular pseudo-D-lattice L is an atom if and only if h(a) = 1. As proved in [5], every modular pseudo-D-lattice L of finite height is atomic, i.e. for every b = 0 in L there exists an atom a of L such that a ≤ b.…”

“…2 It is shown in [5] that the centre of a modular complete atomic pseudo D-lattice is atomic. We will need a particular case of this fact, and prove it directly.…”

“…2 Definition 6. 5. We say that a lattice L 0 with least element 0 is atomless if there are no atoms, i.e., for every a ∈ L 0 with a = 0, there exists b ∈ L 0 with 0 < b < a.…”