15 research outputs found
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
Categories of Residuated Lattices
We present dual variants of two algebraic constructions of certain classes of residuated lattices: The Galatos-Raftery construction of Sugihara monoids and their bounded expansions, and the Aguzzoli-Flaminio-Ugolini quadruples construction of srDL-algebras. Our dual presentation of these constructions is facilitated by both new algebraic results, and new duality-theoretic tools. On the algebraic front, we provide a complete description of implications among nontrivial distribution properties in the context of lattice-ordered structures equipped with a residuated binary operation. We also offer some new results about forbidden configurations in lattices endowed with an order-reversing involution. On the duality-theoretic front, we present new results on extended Priestley duality in which the ternary relation dualizing a residuated multiplication may be viewed as the graph of a partial function. We also present a new Esakia-like duality for Sugihara monoids in the spirit of Dunn\u27s binary Kripke-style semantics for the relevance logic R-mingle
Distributive Laws in Residuated Binars
In residuated binars there are six non-obvious distributivity identities of ⋅,/,∖ over ∧,∨. 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
Residuation algebras with functional duals
We employ the theory of canonical extensions to study residuation algebras whose associated relational structures are functional, i.e., for which the ternary relations associated to the expanded operations admit an interpretation as (possibly partial) functions. Providing a partial answer to a question of Gehrke, we demonstrate that functionality is not definable in the language of residuation algebras (or even residuated lattices), in the sense that no equational or quasi-equational condition in the language of residuation algebras is equivalent to the functionality of the associated relational structures. Finally, we show that the class of Boolean residuation algebras such that the atom structures of their canonical extensions are functional generates the variety of Boolean residuation algebras
Distributive laws in residuated binars
In residuated binars there are six non-obvious distributivity identities of
,, over . 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
Residuation algebras with functional duals
We employ the theory of canonical extensions to study residuation algebras
whose associated relational structures are functional, i.e., for which the
ternary relations associated to the expanded operations admit an interpretation
as (possibly partial) functions. Providing a partial answer to a question of
Gehrke, we demonstrate that no universal first-order sentence in the language
of residuation algebras is equivalent to the functionality of the associated
relational structures
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
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
Some modal and temporal translations of generalized basic logic
We introduce a family of modal expansions of Łukasiewicz 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 Łukasiewicz 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 Gödel-McKinsey-Tarski translation. The first of these interprets generalized basic logic in a modal Łukasiewicz logic in the spirit of the classical modal logic S4, whereas the second interprets generalized basic logic in a temporal variant of the latter
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