In this work we study the notions of structural and universal completeness both from the algebraic and logical point of view. In particular, we provide new algebraic characterizations of quasivarieties that are actively and passively universally complete, and passively structurally complete. We apply these general results to varieties of bounded lattices and to quasivarieties related to substructural logics. In particular we show that a substructural logic satisfying weakening is passively structurally complete if and only if every classical contradiction is explosive in it. Moreover, we fully characterize the passively structurally complete varieties of MTL-algebras, i.e., bounded commutative integral residuated lattices generated by chains.

The present paper investigates proof-theoretical and algebraic properties for the probability logic FP(Ł,Ł), meant for reasoning on the uncertainty of Łukasiewicz events. Methodologically speaking, we will consider a translation function between formulas of FP(Ł,Ł) to the propositional language of Łukasiewicz logic that allows us to apply the latter and the well-developed theory of MV-algebras directly to probabilistic reasoning. More precisely, leveraging on such translation map, we will show proof-theoretical properties for FP(Ł,Ł) and introduce a class of algebras with respect to which FP(Ł,Ł) will be proved to be locally sound and complete. Finally, we will apply these previous results to investigate what we called “probabilistic unification problem”. In this respect, we will prove that Ghilardi's algebraic view on unification can be extended to our case and, on par with the Łukasiewicz propositional case, we show that probabilistic unification is of nullary type.

Abstract We show that the algebraic category of finitely presented Wajsberg hoops is equivalent to a non-full subcategory of finitely presented MV-algebras. We use this connection to show how methods and techniques developed to study MV-algebras can be adapted to study Wajsberg hoops, as well. In particular, we show that finitely presented Wajsberg hoops are dually equivalent to a subcategory of rational polyhedra with $\mathbb {Z}$-maps. We use the duality to provide a geometrical characterization of finitely generated projective and exact Wajsberg hoops. As applications, we study logical properties of the $0$-free fragment of Łukasiewicz logic, seen as a substructural logic. We show that, while deducibility in the fragment is equivalent to deducibility among 0-free formulas in Łukasiewicz logic, the same is not true for the admissibility of rules: there are rules written in the $0$-free language that are admissible in Wajsberg hoops but not in MV-algebras. Moreover, we show that the unification type of Wajsberg hoops is nullary, while the exact unification type is unitary, therefore showing decidability of admissible rules in the fragment.

For functional representation in an algebraizable logic we mean a representation of the algebras of formulas of the logic by means of (possibly real-valued) functions. Functional representations have shown to be a key tool for the study of non-classical logics, since they allow to regard formulas as functions and, by means of them, to approach the study of typical proof theoretical properties of the logics by means of their functional semantics. In the realm of (algebraizable) fuzzy logics, possibly the most well-known result in this respect is McNaughton theorem that shows formulas of the infinite-valued Lukasiewicz calculus to correspond, up to logical equivalence, to real valued continuous and piecewise linear functions. The functional representation for many-valued logics has been very recently shown in a paper by the second author to have an impact outside the purely logical realm and, in particular, they can be applied to study properties of artificial neural networks. In this contribution, we will provide a functional representation for the probability modal logic FP(L) that builds on Lukasiewicz calculus by adding to it a unary operator P that reads “it is probable that”. While the logic FP(L) is not algebraizable, at least not in the usual sense due to Blok and Pigozzi, we still can provide a functional representation result for its modal formulas. In order to do so, we adapt the usual universal algebraic methods to this peculiar setting, and moreover we make use of some techniques developed in a recent paper by two of the authors where a class of purely algebraic models for FP(L) based on de Finetti's coherence criterion have been introduced and studied. Our contribution will present two ways of providing a functional representation of the algebras of formulas of the modal logic FP(L): a local one, that relies on de Finetti's coherence argument; and a global one that, instead, relies on probability distributions on a finite domain.

We develop a unifying approach to study projectivity and unification in substructural logics corresponding to varieties of residuated lattices generated by generalized rotation constructions. These include many interesting varieties especially in the realm of mathematical fuzzy logics. Our main results pertain what we shall call radical-determined varieties of rotations, which include all of the most relevant varieties in this framework. We characterize free algebras in a radical-determined variety of rotations in terms of weak Boolean products of rotations of free algebras in the variety of radicals, the latter being the intersections of maximal filters of the algebras in . Then we use such description to study projectivity in these varieties of rotations, characterizing finitely generated projective algebras. Moreover, we show that the strong unitary unification type of a variety of radicals implies the strong unitary type for the generated variety of rotations, which can be used to deduce the decidability of the admissibility of rules. As relevant applications of our general results, we obtain that product logic and nilpotent minimum logic have (strong) unitary unification type.

Abstract We study (strictly) join irreducible varieties in the lattice of subvarieties of residuated lattices. We explore the connections with well-connected algebras and suitable generalizations, focusing in particular on representable varieties. Moreover, we find weakened notions of Halldén completeness that characterize join irreducibility. We characterize strictly join irreducible varieties of basic hoops and use the generalized rotation construction to find strictly join irreducible varieties in subvarieties of $\mathsf{MTL}$-algebras. We also obtain some general results about linear varieties of residuated lattices, with a particular focus on representable varieties, and a characterization for linear varieties of basic hoops.

Abstract In this paper, we use the generalize d rotation construction to lift results from the lattice of subvarieties of basic hoops to some parts of the lattice of subvarieties of monoidal t-norm based logic-algebras. In particular, we study splitting algebras for (the lattice of subvarieties of) varieties generated by generalized rotations of basic hoops and relevant subvarieties such as Wajsberg hoops, cancellative hoops and Gödel hoops. Finally, we show that the generalized rotation construction preserves the amalgamation property.

This article studies the class of strongly perfect MTL-algebras, i.e. MTL-algebras having an involutive co-radical, and the variety they generate, namely |$\mathbb{SBP}_0$|⁠. Once these structures will be introduced, we will first establish categorical equivalences for several of their relevant proper subvarieties by employing a generalized notion of triplets whose main components are a Boolean algebra and a prelinear semihoop. When triplets are further expanded by a suitable operation between their semihoop reducts, we define a category of quadruples that are equivalent to the whole category of SBP|$_0$|-algebras. Finally, we will provide an explicit representation of SBP|$_0$|-algebras in terms of (weak) Boolean products.