Perfect GMV-Algebras
Antonio Di Nola, Anatolij Dvurečenskij, Constantine Tsinakis · Communications in Algebra · 2008
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.