Abstract
This article provides an exposition of the main results presented in our book “Four Functors: the ‘ancestor’ of Mundici’s equivalence between unital commutative l-groups and MV algebras” appeared in autumn of 2025.
Daniele Mundici has proved, in 1986, that the category of unital commutative l-groups is equivalent to the category of MV algebras, by proving, in one step, the existence of the two quasi-inverse functors Γ and Ξ.
Marco Abbadini has presented in 2021 his equivalence ´a la Mundici for commutative l-monoids, in two steps, with four functors. Mundici’s equivalence itself can be proved in two steps, with four functors, by introducing the intermediary category of positive-unital commutative l-groups.
We have presented in our book the following ‘ancestor’ of Mundici’s equivalence, by Abbadini’s ‘style’ (i.e. in two steps, with four func- tors): The category of strong m0-ME structures is not equivalent to the (intermediary) category of strong m0-ME positive cones, but the category of strong m0-ME positive cones is equivalent to the category of XY algebras.