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Lexicographic effect algebras. (English) Zbl 1350.03046

An effect algebra (with Riesz decomposition property, RDP) represented as an interval \([0,u]\) in the abelian unital po-group \(G\) with interpolation is denoted by \(\Gamma(G,u)\). In the paper, a lexicographic effect algebra is defined to be an interval \(\Gamma(H \buildrel\rightarrow\over\times G, (u,0))\), where \((H,u)\) is a unital and \(G\) is a directed po-group, both abelian, while the multiplication means the operation of forming the lexicographic product of po-groups. Conditions are studied under which an effect algebra is of this form. Various representation theorems are proved, in particular for the so called strongly \((H,u)\)-perfect effect algebras (in the form of lexicographic product) and for lexicographic effect algebras with RDP (as a subdirect product of special lexicographic ideals with RDP). Also, the equivalence of the categories of strongly \((H,u)\)-perfect effect algebras with RDP and of directed po-groups with interpolation is proved.

MSC:

03G12 Quantum logic
06D35 MV-algebras
06F15 Ordered groups
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References:

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