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The Loomis-Sikorski theorem for EMV-algebras. (English) Zbl 1439.06007

The classical theorem of Loomis-Sikorski says that any \(\sigma\)-complete Boolean algebra is the \(\sigma\)-homomorphic image of a \(\sigma\)-algebra of sets. This theorem was generalized for MV-algebras: Any \(\sigma\)-complete MV-algebra is the \(\sigma\)-homomorphic image of a \(\sigma\)-tribe (of [0,1]-valued functions). Recently, the authors [Fuzzy Sets Syst. 373, 116–148 (2019; Zbl 1423.06048)] introduced a new structure which they called EMV-algebra, an algebraic extension of an MV-algebra not necessarily with a top element. It is shown that any \(\sigma\)-complete EMV-algebra is the \(\sigma\)-homomorphic image of a EMV-tribe of fuzzy sets. To show this, the authors “introduce the hull-kernel topology of the maximal ideals of EMV-algebras and the weak topology of state-morphisms which are EMV-homomorphisms from the EMV-algebra into the MV-algebra of the real interval [0,1]”.
Reviewer: Hans Weber (Udine)

MSC:

06D35 MV-algebras
06B23 Complete lattices, completions
06B30 Topological lattices

Citations:

Zbl 1423.06048
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References:

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