TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 365, Number 3, March 2013, Pages 1219–1249 S 0002-9947(2012)05573-5 Article electronically published on October 31, 2012
RESIDUATED FRAMES WITH APPLICATIONS TO DECIDABILITY
NIKOLAOS GALATOS AND PETER JIPSEN
Abstract. Residuated frames provide relational semantics for substructural logics and are a natural generalization of Kripke frames in intuitionistic and modal logic, and of phase spaces in linear logic. We explore the connection between Gentzen systems and residuated frames and illustrate how frames provide a uniform treatment for semantic proofs of cut-elimination, the fi- nite model property and the finite embeddability property, which imply the decidability of the equational/universal theories of the associated residuated lattice-ordered groupoids. In particular these techniques allow us to prove that the variety of involutive FL-algebras and several related varieties have the finite model property.
Substructural logics and their algebraic formulation as varieties of residuated lattices and FL-algebras provide a general framework for a wide range of logical and algebraic systems, such as Classical propositional logic ↔ Boolean algebras Intuitionistic logic ↔ Heyting algebras Lukasiewicz logic ↔ MV-algebras Abelian logic ↔ abelian lattice-ordered groups Basic fuzzy logic ↔ BL-algebras Monoidal t-norm logic ↔ MTL-algebras Noncommutative mult. add. linear logic ↔ InFL-algebras Full Lambek calculus ↔ FL-algebras and lattice-ordered groups, symmetric relation algebras and many other systems. In this paper we introduce residuated frames and show that they provide rela- tional semantics for substructural logics and representations for residuated struc- tures. Our approach is driven by the applications of the theory. As is the case with Kripke frames for modal logics, residuated frames provide a valuable tool for solving both algebraic and logical problems. Moreover we show that there is a direct link between Gentzen-style sequent calculi and our residuated frames, which gives insight into the connection between a cut-free proof system and the finite embeddability property for the corresponding variety of algebras. After an overview of residuated structures and certain types of closure opera- tors called nuclei, we define residuated frames and Gentzen frames (Section 2), and provide several examples. We then prove a general homomorphism theorem in the
Received by the editors August 29, 2008 and, in revised form, February 22, 2011. 2010 Mathematics Subject Classification. Primary 06F05; Secondary 08B15, 03B47, 03G10, 03F05. Key words and phrases. Substructural logic, Gentzen system, residuated lattice, residuated frame, cut elimination, decidability, finite model property, finite embeddability property.