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From implicative reducts to Mundici’s functor

The connection between substructural logics and residuated lattices is one of the most relevant results of algebraic logic. Indeed, it establishes a framework where different systems, or equivalently, classes of structures, can be both compared and studied uniformly. Among the most well-known connections among different structures in this framework surely stands Mundici’s theorem, which establishes a categorical equivalence between the algebraic category of MV-algebras and lattice-ordered abelian groups (abelian l-groups in what follows) with strong order unit (an archimedean element with respect to the lattice order), with unit preserving homomorphisms. This equivalence, connecting the equivalent algebraic semantics of infinite-valued Lukasiewicz logic (i.e., MV-algebras) with ordered groups, has been deeply investigated and also extended to more general structures.

Alternative algebraic approaches to Mundici’s functor have been proposed by other authors. In the present contribution we re-elaborate Rump’s work, which is inspired by Bosbach’s idea, and focuses on structures with only one implication and a constant (whereas Bosbach’s cone algebras have two implications). The key idea is to characterize which structures in this reduced signature embed in an l-group. We find conditions that are different (albeit equivalent) to the ones found by Rump, and moreover we extend some of Rump’s constructions to categorical equivalences of the algebraic categories involved.

Valeria Giustarini is a Master Student at the Department of Information Engineering and Mathematics, University of Siena.

This is a specialized seminar organized by the Logic department. If you want to participate in this seminar, please contact with Tommaso Flaminio <>

The seminar has two parts. The first will be from 10:00 to 12:00 and the second from 14:00 to 16:00.