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When the lexicographic product of two po-groups has the Riesz decomposition property. (English) Zbl 1420.06027

Summary: We study conditions when a certain type of the Riesz Decomposition Property (RDP for short) holds in the lexicographic product of two po-groups. Defining two important properties of po-groups, we extend known situations showing that the lexicographic product satisfies RDP or even \(\mathrm {RDP}_1\), a stronger type of RDP. We recall that a very strong type of RDP, \(\mathrm {RDP}_2\), entails that the group is lattice ordered. RDP’s of the lexicographic products are important for the study of lexicographic pseudo effect algebras, or perfect types of pseudo MV-algebras and pseudo effect algebras, where infinitesimal elements play an important role both for algebras as well as for the first order logic of valid but not provable formulas.

MSC:

06F15 Ordered groups
06D35 MV-algebras
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
03G12 Quantum logic
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