Abstract
We present a survey on pseudo-effect algebras and pseudo MV-algebras, which generalize effect algebras and MV-algebras by dropping the assumption on commutativity. A non-commutative logic is nowadays used even in programming languages. We show when a pseudo-effect algebra E is an interval in a unital po-group. This is possible, e.g. if E satisfies a Riesz-type decomposition property, i.e. another kind of distributivity with respect to addition. Every pseudo MV-algebra is an interval in a unital ℓ-group. We study a case when compatibility can be expressed by a pseudo MV-structure, i.e. when E can be covered by blocks being pseudo MV-algebras. Finally, we study the state space of such structures.
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Dvurečenskij, A., Vetterlein, T. Non-Commutative Algebras and Quantum Structures. International Journal of Theoretical Physics 43, 1599–1612 (2004). https://doi.org/10.1023/B:IJTP.0000048806.69906.11
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DOI: https://doi.org/10.1023/B:IJTP.0000048806.69906.11