Most of my research fits into the broad category of algebraic logic or logically motivated algebra. Below I briefly describe some of the research topics I have worked on to provide a quick glimpse at my interests (mainly to other researchers in the field of algebraic logic).
Algebras of fractions and bimonoids (with Nick Galatos)
A well-known theorem due essentially to Steinitz (1910) states that each cancellative commutative monoid can be embedded into an Abelian group of fractions, where each element has the form a b-1 for some a, b in the original monoid. A more recent result due to Bahls et al. (2003) extends this to obtain lattice-ordered groups of fractions of certain cancellative residuated lattices. In particular, the negative cones of lattice-ordered groups (consisting of the elements below the multiplicative unit) form a variety of residuated lattices, and each algebra in this variety can be embedded into a lattice-ordered group of fractions. Moreover, the group of fractions functor and the negative cone functor yield a categorical equivalence between the two varieties of residuated lattices (unlike in the case of groups of fractions of cancellative commutative monoids). In particular, each lattice-ordered group can be reconstructed up to isomorphism from its negative cone as its lattice-ordered group of fractions.
We extend the group of fractions construction to a wider class of ordered algebras that we call bimonoids, which allows us to view involutive residuated structures other than merely groups or lattice-ordered groups as algebras of fractions. For example, we show that an algebra of fractions may be constructed for each Brouwerian algebra. (These are Heyting algebras without the assumption that a bottom element exists.) This again yields a categorical equivalence between the variety of Brouwerian algebras and a variety of commutative idempotent involutive residuated lattices given by the algebra of fractions functor and the negative cone functor. In particular, Brouwerian algebras are precisely the negative cones of commutative idempotent involutive residuated lattices. Since Brouwerian algebras are much better understood than commutative idempotent involutive residuated lattices, this equivalence can be used to shed more light on the latter variety.
Inconsistency lemmas in abstract algebraic logic (with Tomáš Lávička)
An important part of the field of abstract algebraic logic is the study of a very general hierarchy of deduction theorems (among others parametrized, local, and contextual deduction theorems) generalizing the well-known deduction theorems of classical and intuitionistic logic to e.g. substructural logics and global modal logics. Standard results state that under some mild assumptions having a certain form of the deduction theorem (a syntactic property) is equivalent to some semantic property (such as the filter extension property).
More recently, logics which enjoy a so-called (global) inconsistency lemma relating inconsistency and derivability were studied in an analogical manner by James Raftery. Extending his work, we build a hierarchy of inconsistency lemmas parallel to the hierarchy of deduction theorems. Some interesting new phenomena occur in this hierarchy, for example we show that under some mild assumptions the so-called dual local inconsistency lemma is the syntactic counterpart of semisimplicity.
Exploring the landscape of super-Belnap logics (partly with Hugo Albuquerque and Umberto Rivieccio)
The four-valued Belnap–Dunn logic is a well-known and well-motivated logic for handling inconsistent and incomplete information. However, until recently very little was known about the extensions of this logic (stronger logics in the same language) apart from its three-valued cousins, namely the strong Kleene logic, Priest's Logic of Paradox, and Kleene's logic of order. Umberto Rivieccio, who initiated the study of these logics, dubbed them super-Belnap logics. Following his proposal to study this family in more detail, I provided a basic map of the landscape of super-Belnap logics and identified those which satisfy some natural metalogical properties. Remarkably, it turns out that the lattice of finitary super-Belnap logics can be described in graph-theoretic terms. In fact, results from graph theory can be exploited to prove non-trivial results about super-Belnap logics such as the existence of a non-finitary super-Belnap logic.
Mapping the expansions of the Belnap–Dunn logic
Many different expansions of the four-valued Belnap–Dunn logic (i.e. logics defined over the same four truth values but with different sets of connectives) have been introduced by various researchers. However, a synoptic view of these which would allow us to easily place a given expansion on the map of all possible expansions is still largely missing. This would involve understanding the structure of the lattice of four-valued clones (i.e. of algebras on the four-element set up to termwise equivalence) which contain the connectives of the Belnap–Dunn logic and locating the logics determined by these clones within the various hierarchies of abstract algebraic logic (which classify logics according to their intrinsic language-independent properties).
As a contribution in this direction I have characterized various natural four-valued clones (partly overlapping with recent work of Ofer Arieli and Arnon Avron) and classified the logics of all clones which contain the basic operations of the Belnap–Dunn logic in the hierarchies of abstract algebraic logic. For example, it turns out that within this family the classes of protoalgebraic and equivalential logics coincide.