A CATEGORICAL-ALGEBRAIC EXPLORATION OF MODELS FOR MANY-VALUED LOGIC

In this thesis, we study from a categorical-algebraic point of view the structures of lattice-ordered groups and MV-algebras, which are used in modeling logic with many truth values. In the first part, we show that the semi-abelian category of lattice-ordered groups satisfies several important properties, namely it is fiber-wise algebraically cartesian closed, strongly protomodular, and its full subcategory of lattice-ordered abelian groups is algebraically coherent. In the second part, we observe that the category of MV-algebras is protomodular and arithmetical, and we pursue a thorough exploration of the idempotent elements of an MV-algebra. These elements allow to build a right adjoint to the inclusion of Boolean algebras into MV-algebras. This adjoint is used to construct the Pierce spectrum of an MV-algebra, which has very similar properties to the Pierce spectrum of a unitary ring. In the third part, we investigate the relationship between the category of MV-algebras and its two full subcategories of perfect and semisimple algebras. We show that this pair of subcategories defines a pretorsion theory, which is the generalization to a non-pointed context of the classical notion of torsion 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. In the last part of the thesis, we use the case of MV-algebras as a guideline to develop a non-pointed version of the theory of protoadditive functors. In particular, we show that, under some mild assumptions, there is a one-to-one correspondence between pretorsion theories and stable factorization systems, and we connect such pretorsion theories to the Galois structures determined by the reflections onto the torsion-free parts. Besides MV-algebras, interesting examples of this situation can be found in M-sets, Heyting algebras, and simplicial sets.