arXiv subjects

Wesley Fussner

Publications and source records attributed to Wesley Fussner.

At least 19 recordsLinked to original sources

arXiv2025

Maehara Interpolation in Extensions of R-mingle

We show that there are exactly five quasivarieties of Sugihara algebras with the amalgamation property, and that all of these have the relative congruence extension property. As a consequence, we obtain that the amalgamation property and transferable injections property coincide for arbitrary quasivarieties of Sugihara algebras. These results provide a complete description of arbitrary (not merely axiomatic) extensions of the logic R-mingle that have the Maehara interpolation property, and further demonstrates that the Robinson property and Maehara interpolation property coincide for arbitrary extensions of R-mingle. Further, we show that the question of whether a given finitely based extension of R-mingle has the Maehara interpolation property is decidable.

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arXiv2025

Interpolation in Non-Classical Logics

This chapter surveys some of the main results on interpolation in several of the most prominent families of non-classical logics. Special attention is given to the distinction between the two most commonly studied variants of interpolation--namely, Craig interpolation and deductive interpolation. Our discussion focuses primarily on how these properties present in families of logical systems taken as a whole, particularly those comprising all axiomatic extensions of any of several notable non-classical logics. We consider a range of important examples: superintuitionistic and modal logics, fuzzy logics, paraconsistent logics, relevant logics, and substructural logics.

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arXiv2025

Revisiting Interpolation in Relevant Logics

There are exactly two maximal schematic extensions of the relevant logic R with the variable sharing property. We establish that one of them has a strong form of interpolation for deducibility, thereby giving an example of a well-known relevant logic with interpolation.

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arXiv2025

Agent Interpolation for Knowledge

We define a new type of proof formalism for multi-agent modal logics with S5-type modalities. This novel formalism combines the features of hypersequents to represent S5 modalities with nested sequents to represent the T-like modality alternations. We show that the calculus is sound and complete, cut-free, and terminating and yields decidability and the finite model property for multi-agent S5. We also use it to prove the Lyndon (and hence Craig) interpolation property for multi-agent S5, considering not only propositional atoms but also agents to be part of the common language. Finally, we discuss the difficulties on the way to extending these results to the logic of distributed knowledge and to deductive interpolation.

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arXiv2025

Algebraic Proof Theory for Infinitary Action Logic

We exhibit a uniform method for obtaining (wellfounded and non-wellfounded) cut-free sequent-style proof systems that are sound and complete for various classes of action algebras, i.e., Kleene algebras enriched with meets and residuals. Our method applies to any class of *-continuous action algebras that is defined, relative to the class of all *-continuous action algebras, by analytic quasiequations. The latter make up an expansive class of conditions encompassing the algebraic analogues of most well-known structural rules. These results are achieved by wedding existing work on non-wellfounded proof theory for action algebras with tools from algebraic proof theory.

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arXiv2024

Amalgamation in Semilinear Residuated Lattices

We survey the state of the art on amalgamation in varieties of semilinear residuated lattices. Our discussion emphasizes two prominent cases from which much insight into the general picture may be gleaned: idempotent varieties and their generalizations ($n$-potent varieties, knotted varieties), and cancellative varieties and their relatives (MV-algebras, BL-algebras). Along the way, we illustrate how general-purpose tools developed to study amalgamation can be brought to bear in these contexts and solve some of the remaining open questions concerning amalgamation in semilinear varieties. Among other things, we show that the variety of commutative semilinear residuated lattices does not have the amalgamation property. Taken as a whole, we see that amalgamation is well understood in most interesting varieties of semilinear residuated lattices, with the last few outstanding open questions remaining principally in the cancellative setting.

math.RA↗

arXiv2024

Interpolation in H\'ajek's Basic Logic

We exhaustively classify varieties of BL-algebras with the amalgamation property, showing that there are only countably many of them and solving an open problem of Montagna. As a consequence of this classification, we obtain a complete description of which axiomatic extensions of H\'{a}jek's basic fuzzy logic BL have the deductive interpolation property. Along the way, we provide similar classifications of varieties of basic hoops with the amalgamation property and axiomatic extensions of the negation-free fragment of BL with the deductive interpolation property.

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arXiv2023

Interpolation and the Exchange Rule

It was proved by Maksimova in 1977 that exactly eight varieties of Heyting algebras have the amalgamation property, and hence exactly eight axiomatic extensions of intuitionistic propositional logic have the deductive interpolation property. The prevalence of the deductive interpolation property for axiomatic extensions of substructural logics and the amalgamation property for varieties of pointed residuated lattices, their equivalent algebraic semantics, is far less well understood, however. Taking as our starting point a formulation of intuitionistic propositional logic as the full Lambek calculus with exchange, weakening, and contraction, we investigate the role of the exchange rule--algebraically, the commutativity law--in determining the scope of these properties. First, we show that there are continuum-many varieties of idempotent semilinear residuated lattices that have the amalgamation property and contain non-commutative members, and hence continuum-many axiomatic extensions of the corresponding logic that have the deductive interpolation property in which exchange is not derivable. We then show that, in contrast, exactly sixty varieties of commutative idempotent semilinear residuated lattices have the amalgamation property, and hence exactly sixty axiomatic extensions of the corresponding logic with exchange have the deductive interpolation property. From this latter result, it follows also that there are exactly sixty varieties of commutative idempotent semilinear residuated lattices whose first-order theories have a model completion.

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arXiv2023

Interpolation in Linear Logic and Related Systems

We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that they all enjoy a guarded form of Craig interpolation. We also exhibit continuum-many axiomatic extensions of each of these logics without the deductive interpolation property.

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arXiv2022

Semiconic Idempotent Logic II: Beth Definability and Deductive Interpolation

Semiconic idempotent logic sCI is a common generalization of intuitionistic logic, semilinear idempotent logic sLI, and in particular relevance logic with mingle. We establish the projective Beth definability property and the deductive interpolation property for many extensions of sCI, and identify extensions where these properties fail. We achieve these results by studying the (strong) amalgamation property and the epimorphism-surjectivity property for the corresponding algebraic semantics, viz. semiconic idempotent residuated lattices. Our study is made possible by the structural decomposition of conic idempotent models achieved in the prequel, as well as a detailed analysis of the structure of idempotent residuated chains serving as index sets in this decomposition. Here we study the latter on two levels: as certain enriched Galois connections and as enhanced monoidal preorders. Using this, we show that although conic idempotent residuated lattices do not have the amalgamation property, the natural class of rigid and conjunctive conic idempotent residuated lattices has the strong amalgamation property, and thus has surjective epimorphisms. This extends to the variety generated by rigid and conjunctive conic idempotent residuated lattices, and we establish the (strong) amalgamation and epimorphism-surjectivity properties for several important subvarieties. Using the algebraizability of sCI, this yields the deductive interpolation property and the projective Beth definability property for the corresponding substructural logics extending sCI.

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arXiv2022

Transfer theorems for finitely subdirectly irreducible algebras

We show that under certain conditions, well-studied algebraic properties transfer from the class $\mathcal{Q}_{_\text{RFSI}}$ of the relatively finitely subdirectly irreducible members of a quasivariety $\mathcal{Q}$ to the whole quasivariety, and, in certain cases, back again. First, we prove that if $\mathcal{Q}$ is relatively congruence-distributive, then it has the $\mathcal{Q}$-congruence extension property if and only if $\mathcal{Q}_{_\text{RFSI}}$ has this property. We then prove that if $\mathcal{Q}$ has the $\mathcal{Q}$-congruence extension property and $\mathcal{Q}_{_\text{RFSI}}$ is closed under subalgebras, then $\mathcal{Q}$ has a one-sided amalgamation property (equivalently, for $\mathcal{Q}$, the amalgamation property) if and only if $\mathcal{Q}_{_\text{RFSI}}$ has this property. We also establish similar results for the transferable injections property and strong amalgamation property. For each property considered, we specialize our results to the case where $\mathcal{Q}$ is a variety -- so that $\mathcal{Q}_{_\text{RFSI}}$ is the class of finitely subdirectly irreducible members of $\mathcal{Q}$ and the $\mathcal{Q}$-congruence extension property is the usual congruence extension property -- and prove that when $\mathcal{Q}$ is finitely generated and congruence-distributive, and $\mathcal{Q}_{_\text{RFSI}}$ is closed under subalgebras, possession of the property is decidable. Finally, as a case study, we provide a complete description of the subvarieties of a notable variety of BL-algebras that have the amalgamation property.

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arXiv2021

Mining counterexamples for wide-signature algebras with an Isabelle server

We propose an approach for searching for counterexamples of statements about algebraic structures with a medium-sized signature using the Isabelle proof assistant in an efficient, parallel manner. We contribute a Python client Isabelle server and other scripts implementing our approach, and provide results of our computational experiments. In particular, our experiments yield counterexamples that resolve a previously open question regarding the interdependencies between distributive-like identities in residuated binars.

cs.LO↗

arXiv2021

Some modal and temporal translations of generalized basic logic

We introduce a family of modal expansions of {\L}ukasiewicz logic that are designed to accommodate modal translations of generalized basic logic (as formulated with exchange, weakening, and falsum). We further exhibit algebraic semantics for each logic in this family, in particular showing that all of them are algebraizable in the sense of Blok and Pigozzi. Using this algebraization result and an analysis of congruences in the pertinent varieties, we establish that each of the introduced modal {\L}ukasiewicz logics has a local deduction-detachment theorem. By applying Jipsen and Montagna's poset product construction, we give two translations of generalized basic logic with exchange, weakening, and falsum in the style of the celebrated G\"odel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal {\L}ukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter.

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arXiv2021

Negative Translations of Orthomodular Lattices and Their Logic

We introduce residuated ortholattices as a generalization of -- and environment for the investigation of -- orthomodular lattices. We establish a number of basic algebraic facts regarding these structures, characterize orthomodular lattices as those residuated ortholattices whose residual operation is term-definable in the involutive lattice signature, and demonstrate that residuated ortholattices are the equivalent algebraic semantics of an algebraizable propositional logic. We also show that orthomodular lattices may be interpreted in residuated ortholattices via a translation in the spirit of the double-negation translation of Boolean algebras into Heyting algebras, and conclude with some remarks about decidability.

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arXiv2020

Poset Products as Relational Models

We introduce a relational semantics based on poset products, and provide sufficient conditions guaranteeing its soundness and completeness for various substructural logics. We also demonstrate that our relational semantics unifies and generalizes two semantics already appearing in the literature: Aguzzoli, Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz logic. As a consequence of our general theory, we recover the soundness and completeness results of these prior studies in a uniform fashion, and extend them to infinitely-many other substructural logics.

math.LO↗

arXiv2020

Priestley duality for MV-algebras and beyond

We provide a new perspective on extended Priestley duality for a large class of distributive lattices equipped with binary double quasioperators. Under this approach, non-lattice binary operations are each presented as a pair of partial binary operations on dual spaces. In this enriched environment, equational conditions on the algebraic side of the duality may more often be rendered as first-order conditions on dual spaces. In particular, we specialize our general results to the variety of MV-algebras, obtaining a duality for these in which the equations axiomatizing MV-algebras are dualized as first-order conditions.

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arXiv2019

Distributive laws in residuated binars

In residuated binars there are six non-obvious distributivity identities of $\cdot$,$/$,$\backslash$ over $\wedge, \vee$. We show that in residuated binars with distributive lattice reducts there are some dependencies among these identities; specifically, there are six pairs of identities that imply another one of these identities, and we provide counterexamples to show that no other dependencies exist among these.

math.LO↗

arXiv2018

A topological approach to MTL-algebras

We dualize a construction of Aguzzoli-Flaminio-Ugolini of a large class of MTL-algebras from ordered quadruples consisting of a Boolean algebra, a generalized MTL-algebra, and two maps parameterizing the connection between these pieces. Our dualized construction gives a uniform way of building the extended Priestley duals of MTL-algebras in this class from the Stone duals of their Boolean skeletons, the extended Priestley duals of their radicals, and a family of closure operators associating the two. In order to facilitate this work, we also offer some new results regarding the extended Priestley duals of MTL-algebras and GMTL-algebras.

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