Locality in Residuated-Lattice Structures

Many-valued models generalise the structures from classical model theory by defining truth values for a model with an arbitrary algebra. Just as algebraic varieties provide semantics for many non-classical propositional logics, models defined over algebras in a variety provide the semantics for the corresponding non-classical predicate logics. In particular, models defined over varieties of residuated lattices represent the model theory for first-order substructural logics. In this paper we study the extent to which the classical locality theorems from Hanf and Gaifman hold true in the residuated lattice setting. We demonstrate that the answer is sensitive both to how locality is understood in the generalised context and the behaviour of the truth-defining algebra. In the case of Hanf’s theorem, we will show that the theorem fails for the natural understanding of local neighbourhoods, but is recoverable with an alternative understanding for well-connected residuated lattices. For Gaifman’s theorem, rather than consider Gaifman normal forms directly we focus on the main lemma of the theorem from textbook proofs - that models which satisfy the same basic local sentences are elementarily equivalent. We prove that for a number of different understandings of locality, provided the algebra is well-behaved enough to express locality in its syntax, this main lemma can be recovered. In each case we will see the importance of an order-interpreting connective which creates a link between the modelling relation for models and formulas and the valuation function from formulas into the algebra. This link enables a syntactic encoding of back-and-forth systems providing the main technical ingredient to proofs of the main locality results.

💡 Research Summary

This paper investigates how the classical locality theorems of Hanf and Gaifman extend to many‑valued model theory when truth values are taken from a residuated lattice. The authors begin by recalling that in classical first‑order logic, locality results are powerful tools: Hanf locality links the number of isomorphic local neighbourhoods to equivalence of sentences of bounded quantifier rank, while Gaifman locality states that every formula is equivalent to a Boolean combination of formulas whose quantifiers range only over a bounded neighbourhood in the Gaifman graph.

The study adopts residuated lattices as the algebraic semantics for a wide family of substructural logics (including Boolean, Heyting, MV‑algebras, and lattice‑ordered groups). A residuated lattice A is a lattice equipped with a monoid operation “·” and two residuals “/” and “\” satisfying the usual adjunction property. Crucially, the residuals provide an order‑interpreting connective τ(x,y) (often the implication →) that is true exactly when x ≤ y in the lattice order. This connective creates a tight link between the syntactic satisfaction relation ⊨ and the valuation function ‖·‖_M that maps formulas to elements of A.

Hanf locality. The authors first try to transplant the standard definition of a radius‑k neighbourhood (based on the Gaifman graph) to the many‑valued setting. They show that, without further constraints on the algebra, the Hanf theorem fails: two structures may have the same number of isomorphic k‑neighbourhoods yet assign different truth values to sentences of the same quantifier rank. The failure is illustrated with infinite chains and with algebras lacking a top or bottom element.

However, when the underlying lattice is well‑connected (i.e., for all a,b we have a∨b≥1 iff a≥1 or b≥1) and possesses a distinguished element just below the unit (a co‑atom) or a global lower bound, a modified notion of neighbourhood can be introduced. This notion restricts attention to elements whose truth values are “close to 1”. Under this refined definition the Hanf theorem is recovered: the number of isomorphic modified neighbourhoods determines equivalence of sentences up to a given quantifier rank. The proof proceeds by defining r‑types (sets of formulas of quantifier rank ≤ r) and showing that, in a well‑connected lattice, there are only finitely many r‑types up to algebraic equivalence; equality of neighbourhood counts forces equality of r‑types, yielding Hanf locality.

Gaifman locality. Rather than attempting to construct Gaifman normal forms directly, the paper focuses on the central lemma used in textbook proofs: if two structures satisfy the same basic local sentences, then they are elementarily equivalent. The authors identify several “understandings” of locality (different ways to bound the radius, different ways to encode distance) and show that, provided the algebra can express the order‑interpreting connective and contains either a lower bound or a co‑atom, the lemma can be proved.

The key technical device is a syntactic encoding of a back‑and‑forth system. Given two A‑structures M and N, a partial map f: M → N is built step‑by‑step. At each step the map must preserve the truth of τ‑formulas τ(a,b) (which encode a ≤ b). Because τ is definable in the language, the preservation condition can be expressed as a set of first‑order sentences. The construction guarantees that every basic local sentence true in M remains true in N and vice versa, which yields a partial isomorphism that can be extended to a full elementary equivalence. Thus the Gaifman lemma holds for any residuated lattice that is well‑connected and either bounded below or possesses a co‑atom.

Comparison with semiring semantics. The authors compare their results with earlier work on ordered semirings, where only conjunction and disjunction are generalized and no residual (hence no order‑interpreting connective) exists. In that setting, both Hanf and Gaifman results are more fragile; the lack of τ prevents a syntactic back‑and‑forth encoding. The presence of residuals in residuated lattices therefore makes the locality theory substantially richer.

Applications and outlook. The paper sketches how the recovered locality lemmas can be used to prove non‑definability of certain database queries in many‑valued logics, exactly as in the classical case. It also outlines future research directions: extending the results to infinite structures, investigating 0‑1 laws for residuated lattices, exploring locality in non‑well‑connected algebras, and developing full Gaifman normal forms for substructural logics.

In summary, the work demonstrates that classical locality theorems are not lost in the transition to many‑valued model theory; rather, their validity hinges on specific algebraic properties of the residuated lattice, most notably the existence of an order‑interpreting connective and suitable bounds (well‑connectedness, co‑atoms, or global minima). When these conditions are met, both Hanf and Gaifman locality can be recovered, providing a solid foundation for further logical and computational investigations in substructural and fuzzy logics.