Decomposition of effect algebras and the Hammer–Sobczyk theorem

Abstract

We prove an algebraic and a topological decomposition theorem for complete D-lattices (i.e., lattice-ordered effect algebras). As a consequence, we obtain a HammerSobczyk type decomposition theorem for modular measures on D-lattices.

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

“…Finally, thanks to the algebraic and topological decompositions, we obtain a HammerSobczyk decomposition theorem for exhaustive modular measures on pseudo-D-lattices (see Theorem 7.8), which extends the results of [2,7] and, as a consequence, a compactness result concerning R n -valued modular measures (see Theorem 7.13).…”

“…In order to establish the algebraic decomposition, we could not follow the lines of our previous paper [2], due to the lack of commutativity. A remarkable achievement has been to recover a sort of symmetry in some particular cases.…”

“…This result has been generalized in [25] for group-valued measures on Boolean algebras using topological methods. Afterwards, the result has been extended to MV-algebras [7] and to D-lattices [2], still employing topological methods.…”

An extension theorem for modular measures on effect algebras

“…Observe that (L, U ) is complete by [31, 1. We continue by recalling the following theorem which is contained in [5]. We can refine the previous theorem in the following way: P r o o f. We prove only the convexity assertion in (b).…”

Lattice uniformities on pseudo-D-lattices

“…Our aim in studying modular measures is to extend to pseudo-D-lattices the results we have found in D-lattices (see [3,5,7] and many others), which allowed us to develop a common generalization of topological methods that have been employed both in Noncommutative Measure Theory and in Fuzzy Measure Theory. In this perspective the present article is an important step inasmuch as here we prove some basic facts to be used in subsequent papers.…”