Maximal Subsets of Pairwise Summable Elements in Generalized Effect Algebras

Abstract

Abstract. We show that in any generalized effect algebra (G; ⊕, 0) a maximal pairwise summable subset is a sub-generalized effect algebra of (G; ⊕, 0), called a summability block. If G is lattice ordered, then every summability block in G is a generalized MV-effect algebra. Moreover, if every element of G has an infinite isotropic index, then G is covered by its summability blocks, which are generalized MV-effect algebras in the case that G is lattice ordered. We also present the relations between summability blocks and compatibility blocks of G. Counterexamples, to obtain the required contradictions in some cases, are given.Keywords: (generalized) effect algebra, MV-effect algebra, summability block, compatibility block, linear operators in Hilbert spaces.