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In this paper, we focus on direct limits and inverse limits in the category with generalized pseudo-effect algebras (GPEAs for short) as objects and GPEA-morphisms as morphisms. We show that direct limits exist in the category of GPEAs and direct limits of GPEAs satisfy the Riesz decomposition properties whenever the directed systems of GPEAs satisfy the Riesz decomposition properties. Then, we give a condition under which the quotient of a direct limit of GPEAs is a direct limit of quotients of GPEAs. Moreover, we prove that if inverse systems of GPEAs satisfy the Riesz decomposition properties, then inverse limits also satisfy the Riesz decomposition properties. [ABSTRACT FROM AUTHOR]

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]

Using the unitizations of symmetric-generalized pseudo-effect algebras, we investigate the relationships between the maximal compatible sets in a prelattice symmetric-generalized pseudo-effect algebra and the maximal compatible sets in its unitization. We also prove that a symmetric-generalized pseudo-effect algebra is a symmetric-generalized pseudo MV-effect algebra if and only if its unitization is a symmetric pseudo MV-effect algebra. [ABSTRACT FROM AUTHOR]

We give sufficient and necessary conditions to guarantee that a pseudo-effect algebra admits an ( n + 1)-valued discrete state. We introduce n-perfect pseudo-effect algebras as algebras which can be split into n + 1 comparable slices. We prove that the category of strong n-perfect pseudo-effect algebras is categorically equivalent to the category of torsion-free directed partially ordered groups of a special type. [ABSTRACT FROM AUTHOR]

The definition of E-perfect effect algebras is introduced, and their structure is studied. We study the lexicographical product of an effect algebra with any upwards directed partially ordered Abelian group, and we show that every E-perfect effect algebra is isomorphic with such a kind of the lexicographical product. [ABSTRACT FROM AUTHOR]