A Galois Theory and a Pretorsion Theory in MV-Algebras

Abstract

We explore the relationship between the category of MV-algebras and its full subcategories of perfect and semisimple algebras, showing that this pair of subcategories defines a pretorsion theory. We study the Galois structure associated with the reflection of semisimple MV-algebras, proving that it is admissible from the point of view of categorical Galois theory and characterizing the corresponding central extensions. These central extensions are themselves reflective into the category of surjective MV-algebra morphisms, and the corresponding adjunction is admissible, too. Thanks to this observation, we characterize higher central extensions, and we use them to define a non-pointed version of commutators between ideal subalgebras.