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THE LOOMIS–SIKORSKI THEOREM FOR $EMV$-ALGEBRAS

Part of: Boolean algebras (Boolean rings) Modular lattices, complemented lattices Distributive lattices

Published online by Cambridge University Press:  23 August 2018

ANATOLIJ DVUREČENSKIJ  and
OMID ZAHIRI
ANATOLIJ DVUREČENSKIJ
Affiliation:
Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, SK-814 73 Bratislava, Slovakia Depart. Algebra Geom., Palacký Univer. 17. listopadu 12, CZ-771 46 Olomouc, Czech Republic email dvurecen@mat.savba.sk
OMID ZAHIRI*
Affiliation:
University of Applied Science and Technology, Tehran, Iran email zahiri@protonmail.com
*
✉om.zahiri@gmail.com
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Abstract

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An EMV-algebra resembles an MV-algebra in which a top element is not guaranteed. For $\unicode[STIX]{x1D70E}$-complete $EMV$-algebras, we prove an analogue of the Loomis–Sikorski theorem showing that every $\unicode[STIX]{x1D70E}$-complete $EMV$-algebra is a $\unicode[STIX]{x1D70E}$-homomorphic image of an $EMV$-tribe of fuzzy sets where all algebraic operations are defined by points. To prove it, some topological properties of the state-morphism space and the space of maximal ideals are established.

Keywords

MV-algebraidempotent elementEMV-algebra𝜎-complete EMV-algebraEMV-clanEMV-tribestate-morphismidealfilterhull-kernel topologythe Loomis–Sikorski theorem

MSC classification

Primary: 06C15: Complemented lattices, orthocomplemented lattices and posets
Secondary: 06D35: MV-algebras
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 106 , Issue 2 , April 2019 , pp. 200 - 234
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The first author is grateful for support from grants APVV-16-0073, VEGA no. 2/0069/16 SAV and GAČR 15-15286S.

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