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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]

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]

One kind of generalized measures called quantum measures on finite effect algebras, which fulfil the grade-2 additive sum rule, is considered. One basis of vector space of quantum measures on a finite effect algebra with the Riesz decomposition property (RDP for short) is given. It is proved that any diagonally positive symmetric signed measure $$\lambda $$ on the tensor product $$E\otimes E$$ can determine a quantum measure $$\mu $$ on a finite effect algebra $$E$$ with the RDP such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . Furthermore, some conditions for a grade-2 additive measure $$\mu $$ on a finite effect algebra $$E$$ are provided to guarantee that there exists a unique diagonally positive symmetric signed measure $$\lambda $$ on $$E\otimes E$$ such that $$\mu (x)=\lambda (x\otimes x)$$ for any $$x\in E$$ . At last, it is showed that any grade- $$t$$ quantum measure on a finite effect algebra $$E$$ with the RDP is essentially established by the values on a subset of $$E$$ . [ABSTRACT FROM AUTHOR]

An n-perfect pseudo effect algebra means that it can be decomposed into n+1 comparable slices. We show that such a pseudo effect algebra satisfying a Riesz Decomposition Property type corresponds to the lexicographic product of a cyclic group $\frac{1}{n}\mathbb{Z}$ with some po-group. The analogous result will be proved for strong $\mathbb{Q}$-perfect pseudo effect algebras. [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]