Perfect GMV-Algebras

The focus of this article is the class of perfect GMV-algebras, which includes all noncommutative analogs of perfect MV-algebras. As in the commutative case, we show that each perfect GMV-algebra possesses a single negation, it is generated by its infinitesimal elements, and can be uniquely realized...

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Bibliographic Details
Published in:Communications in algebra 2008-04, Vol.36 (4), p.1221-1249
Main Authors: Nola, A. Di, Dvurečenskij, A., Tsinakis, C.
Format: Article
Language:English
Subjects:
Amalgamation
Categorical equivalence
GMV-algebra
Perfect GMV-algebra
State
Top variety
Unital ℓ-group
Variety
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Summary:The focus of this article is the class of perfect GMV-algebras, which includes all noncommutative analogs of perfect MV-algebras. As in the commutative case, we show that each perfect GMV-algebra possesses a single negation, it is generated by its infinitesimal elements, and can be uniquely realized as an interval in a lexicographical product of the lattice-ordered group of integers and an arbitrary lattice-ordered group. Further, we establish that the category of perfect GMV-algebras is equivalent to the category of all lattice-ordered groups. The variety of GMV-algebras generated by the class of perfect GMV-algebras plays a key role in our considerations. Among other results, we describe a finite equational basis for this variety and prove that it fails to satisfy the amalgamation property. In fact, we show that uncountably many of its subvarieties fail this property.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927870701862852