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The natural question about the sum of observables on σ-complete MV-effect algebras, which was recently defined by A. Dvurečenskij, is how it affects spectra of observables, particularly, their extremal points. We describe boundaries for extremal points of the spectrum of the sum of observables in a general case, and we give necessary and sufficient conditions under which the spectrum attains these boundary values. Moreover, we show that every bounded observable x on a complete MV-effect algebra E can be decomposed into the sum x=x~+x′, where x~ is the greatest sharp observable less than x and x′ is a meager and extremally non-invertible observable. [ABSTRACT FROM AUTHOR]

In this study, we present techniques for constructing a lattice-ordered effect algebra using a given family of MV-effect algebras. First, we introduce the Greechie diagrams of the pastings of a family of MV-effect algebras. As application of the Greechie diagrams, we provide some sufficient conditions for pasting a lattice-ordered effect algebra using a family of MV-effect algebras. Especially, we illustrate a kind of technique of how to obtain a lattice-ordered effect algebra by substituting the atoms of an orthomodular lattice with the lattice-ordered effect algebras. Finally, we prove a version of the “Loop Lemma” for MV-effect algebras pasting. [ABSTRACT FROM AUTHOR]

We discuss the relationships between effect algebras with the Riesz Decomposition Property and partially ordered groups with interpolation. We show that any σ-orthocomplete atomic effect algebra with the Riesz Decomposition Property is an MV-effect algebras, and we apply this result for pseudo-effect algebras and for states. [ABSTRACT FROM AUTHOR]

noncommutative generalizations of effect algebras, we introduce weak commutative pseudoeffect algebras. In this paper, we prove that the generalized pseudoeffect algebras can be unitized if and only if they are weak commutative. Then we discuss the relationships between weak commutative pseudoeffect algebras and weak commutative generalized pseudoeffect algebras. We prove that the category of weak commutative pseudoeffect algebras is a reflective subcategory of weak commutative generalized pseudoeffect algebras. Similarly, we introduce weak commutative pseudodifference posets and show the relationships between weak commutative pseudoeffect algebras and weak commutative pseudodifference posets. [ABSTRACT FROM AUTHOR]