Hopfian $\ell$-groups, MV-algebras and AF C$^*$-algebras

TL;DR

This paper investigates hopfian properties of unital lattice-ordered abelian groups and MV-algebras, applying classification theories to connect these properties with AF C*-algebras and their quotients.

Contribution

It characterizes hopfian and non-hopfian classes of these algebraic structures and links them to AF C*-algebras using K-theory and Elliott classification.

Findings

Identifies classes of hopfian and non-hopfian MV-algebras and lattice-ordered groups.

Connects hopfian properties to AF C*-algebras via K-theory.

Analyzes primitive quotients of Farey-Stern-Brocot AF C*-algebra.

Abstract

An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, including the Behnke-Leptin C$^*$-algebras $\mathcal A_{k,q}$.