Journal of Algebra

A residuated lattice is an ordered algebraic structure such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The collection of all two-sided ideals of a ring forms a lattice upon which one can impose a natural monoid structure making this object into a residuated lattice. Such ideas were investigated by Morgan Ward and R. P. Dilworth in a series of important papers [15, 16, 45–48] and also by Krull in [33]. Since that time, there has been substantial research regarding some specific classes of residuated structures, see for example [1, 9, 26] and [38], but we believe that this is the first time that a general structural theory has been established for the class ℛℒ as a whole. In particular, we develop the notion of a normal subalgebra and show that ℛℒ is an "ideal variety" in the sense that it is an equational class in which congruences correspond to "normal" subalgebras in the same way that ring congruences correspond to ring ideals. As an application of the general theory, we produce an equational basis for the important subvariety ℛℒC that is generated by all residuated chains. In the process, we find that this subclass has some remarkable structural properties that we believe could lead to some important decomposition theorems for its finite members (along the lines of the decompositions provided in [27]).

https://my.vanderbilt.edu/constantinetsinakis/files/2014/03/final-rlpaper-final.pdf

The structure of residuated lattices

Residuation is a fundamental concept of ordered structures and categories. In this survey we consider the consequences of adding a residuated monoid operation to lattices. The resulting residuated lattices have been studied in several branches of mathematics, including the areas of lattice-ordered groups, ideal lattices of rings, linear logic and multi-valued logic. Our exposition aims to cover basic results and current developments, concentrating on the algebraic structure, the lattice of varieties, and decidability.

A Survey of Residuated Lattices

Entailment: The Logic of Relevance and Necessity

This article deals with many-valued modal logics, based only on the necessity operator, over a residuated lattice. We focus on three basic classes, according to the accessibility relation, of Kripke frames: the full class of frames evaluated in the residuated lattice (and so defining the minimum modal logic), the ones evaluated in the idempotent elements and the ones only evaluated in 0 and 1. We show how to expand an axiomatization, with canonical truth-constants in the language, of a finite residuated lattice into one of the modal logic, for each one of the three basic classes of Kripke frames. We also provide axiomatizations for the case of a finite MV chain but this time without canonical truth-constants in the language.

On the Minimum Many-Valued Modal Logic over a Finite Residuated Lattice

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin.We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions.Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples.We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric. This book: Considers both the algebraic and logical perspective within a common framework; Is written by experts in the area; Is easily accessible to graduate students and researchers from other fields; Includes results summarized in tables and diagrams to provide an overview of the area; Is useful as a textbook for a course in algebraic logic, with exercises and suggested research directions; And provides a concise introduction to the subject and leads directly to research topics. The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level.

Residuated Lattices: An Algebraic Glimpse at Substructural Logics

The book is meant to serve two purposes. The first and more obvious one is to present state of the art results in algebraic research into residuated structures related to substructural logics. The second, less obvious but equally important, is to provide a reasonably gentle introduction to algebraic logic. At the beginning, the second objective is predominant. Thus, in the first few chapters the reader will find a primer of universal algebra for logicians, a crash course in nonclassical logics for algebraists, an introduction to residuated structures, an outline of Gentzen-style calculi as well as some titbits of proof theory - the celebrated Hauptsatz, or cut elimination theorem, among them. These lead naturally to a discussion of interconnections between logic and algebra, where we try to demonstrate how they form two sides of the same coin. We envisage that the initial chapters could be used as a textbook for a graduate course, perhaps entitled Algebra and Substructural Logics. As the book progresses the first objective gains predominance over the second. Although the precise point of equilibrium would be difficult to specify, it is safe to say that we enter the technical part with the discussion of various completions of residuated structures. These include Dedekind-McNeille completions and canonical extensions. Completions are used later in investigating several finiteness properties such as the finite model property, generation of varieties by their finite members, and finite embeddability. The algebraic analysis of cut elimination that follows, also takes recourse to completions. Decidability of logics, equational and quasi-equational theories comes next, where we show how proof theoretical methods like cut elimination are preferable for small logics/theories, but semantic tools like Rabin's theorem work better for big ones. Then we turn to Glivenko's theorem, which says that a formula is an intuitionistic tautology if and only if its double negation is a classical one. We generalise it to the substructural setting, identifying for each substructural logic its Glivenko equivalence class with smallest and largest element. This is also where we begin investigating lattices of logics and varieties, rather than particular examples. We continue in this vein by presenting a number of results concerning minimal varieties/maximal logics. A typical theorem there says that for some given well-known variety its subvariety lattice has precisely such-and-such number of minimal members (where values for such-and-such include, but are not limited to, continuum, countably many and two). In the last two chapters we focus on the lattice of varieties corresponding to logics without contraction. In one we prove a negative result: that there are no nontrivial splittings in that variety. In the other, we prove a positive one: that semisimple varieties coincide with discriminator ones. Within the second, more technical part of the book another transition process may be traced. Namely, we begin with logically inclined technicalities and end with algebraically inclined ones. Here, perhaps, algebraic rendering of Glivenko theorems marks the equilibrium point, at least in the sense that finiteness properties, decidability and Glivenko theorems are of clear interest to logicians, whereas semisimplicity and discriminator varieties are universal algebra par exellence. It is for the reader to judge whether we succeeded in weaving these threads into a seamless fabric. - Considers both the algebraic and logical perspective within a common framework. - Written by experts in the area. - Easily accessible to graduate students and researchers from other fields. - Results summarized in tables and diagrams to provide an overview of the area. - Useful as a textbook for a course in algebraic logic, with exercises and suggested research directions. - Provides a concise introduction to the subject and leads directly to research topics. - The ideas from algebra and logic are developed hand-in-hand and the connections are shown in every level.

Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151

https://my.vanderbilt.edu/constantinetsinakis/files/2014/03/gmv040701.pdf

Generalized MV-algebras

We introduce a systematic procedure to transform large classes of (Hilbert) axioms into equivalent inference rules in sequent and hypersequent calculi. This allows for the automated generation of analytic calculi for a wide range of prepositional nonclassical logics including intermediate, fuzzy and substructural logics. Our work encompasses many existing results, allows for the definition of new calculi and contains a uniform semantic proof of cut-elimination for hypersequent calculi.

/pdf/from-axioms-to-analytic-rules-in-nonclassical-logics-5bkbpfobnt.pdf

From Axioms to Analytic Rules in Nonclassical Logics

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 i nc ontr ast 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 − .F urthermo re 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.

/pdf/cancellative-residuated-lattices-28v64gkw2n.pdf

Nikolaos Galatos

Papers

Residuated Lattices: An Algebraic Glimpse at Substructural Logics

Residuated Lattices: An Algebraic Glimpse at Substructural Logics, Volume 151

Generalized MV-algebras

From Axioms to Analytic Rules in Nonclassical Logics

Cancellative residuated lattices