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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 finite model property and the finite embeddability property. We use our results to prove the decidability of the equational and/or universal theory of several varieties of residuated lattice-ordered groupoids, including the variety of involutive FL-algebras.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 Intuitionistic linear logic ↔ ILL-algebras Full Lambek calculus ↔ FL-algebras as well as lattice-ordered groups, symmetric relation algebras and many other systems.In this paper we introduce residuated frames and show that they provide relational semantics for substructural logics and representations for residuated structures. 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.We begin with an overview of residuated structures and certain types of closure operators called nuclei. This leads to the definition of residuated frames (Section 3) and Gentzen frames (Section 4), illustrated by several examples. We then prove a general homomorphism theorem in the setting of Gentzen frames (Thm. 4.2) and Date: August 25, 2008.

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Abstract. Cancellative residuated lattices are natural generalizations of lattice-ordered groups ( -groups). Although cancellative monoids are defined by quasi-equations, the class CanRL of cancellative residuated lattices is a variety. We prove that there are only two commutative subvarieties of CanRL that cover the trivial variety, namely the varieties generated by the integers and the negative integers (with zero). We also construct examples showing that in contrast to -groups, the lattice reducts of cancellative residuated lattices need not be distributive. In fact we prove that every lattice can be embedded in the lattice reduct of a cancellative residuated lattice. Moreover, we show that there exists an order-preserving injection of the lattice of all lattice varieties into the subvariety lattice of CanRL.We define generalized MV-algebras and generalized BL-algebras and prove that the cancellative integral members of these varieties are precisely the negative cones of -groups, hence the latter form a variety, denoted by LG − . Furthermore we prove that the map that sends a subvariety of -groups to the corresponding class of negative cones is a lattice isomorphism from the lattice of subvarieties of LG to the lattice of subvarieties of LG − .Finally, we show how to translate equational bases between corresponding subvarieties, and briefly discuss these results in the context of R. McKenzie's characterization of categorically equivalent varieties.

A generalized BL-algebra (or GBL-algebra for short) is a residuated lattice that satisfies the identities x ∧ y = ((x ∧ y)/y)y = y(y\(x ∧ y)). It is shown that all finite GBL-algebras are commutative, hence they can be constructed by iterating ordinal sums and direct products of Wajsberg hoops. We also observe that the idempotents in a GBL-algebra form a subalgebra of elements that commute with all other elements.Subsequently we construct subdirectly irreducible noncommutative integral GBL-algebras that are not ordinal sums of generalized MV-algebras. We also give equational bases for the varieties generated by such algebras. The construction provides a new way of order-embedding the lattice of -group varieties into the lattice of varieties of integral GBLalgebras.The results of this paper also apply to pseudo-BL algebras.