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Hyper-Archimedean BL-algebras are MV-algebras

✍ Scribed by Esko Turunen

Publisher: John Wiley and Sons
Year: 2007
Tongue: English
Weight: 123 KB
Volume: 53
Category: Article
ISSN: 0044-3050
DOI: 10.1002/malq.200610037

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✦ Synopsis

Abstract

Generalizations of Boolean elements of a BL‐algebra L are studied. By utilizing the MV‐center MV(L) of L, it is reproved that an element x ∈ L is Boolean iff x ∨ x * = 1. L is called semi‐Boolean if for all x ∈ L, x * is Boolean. An MV‐algebra L is semi‐Boolean iff L is a Boolean algebra. A BL‐algebra L is semi‐Boolean iff L is an SBL‐algebra. A BL‐algebra L is called hyper‐Archimedean if for all x ∈ L, x^n^ is Boolean for some finite n ≥ 1. It is proved that hyper‐Archimedean BL‐algebras are MV‐algebras. The study has application in mathematical fuzzy logics whose Lindenbaum algebras are MV‐algebras or BL‐algebras. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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