Peter Jipsen

A Survey 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.

Varieties of Lattices

In this chapter we discuss some of the more recent results and give a general overview of what is currently known about lattice varieties. Of course it is impossible to give a comprehensive account. Often we only cite recent or survey papers, which themselves have many more references. We would like to apologize in advance for any errors, omissions, or miscrediting of results.

Residuated frames with applications to decidability

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

Algebraic aspects of cut elimination

We will give here a purely algebraic proof of the cut elimination theorem for various sequent systems. Our basic idea is to introduce mathematical structures, called Gentzen structures, for a given sequent system without cut, and then to show the completeness of the sequent system without cut with respect to the class of algebras for the sequent system with cut, by using the quasi-completion of these Gentzen structures. It is shown that the quasi-completion is a generalization of the MacNeille completion. Moreover, the finite model property is obtained for many cases, by modifying our completeness proof. This is an algebraic presentation of the proof of the finite model property discussed by Lafont [12] and Okada-Terui [17].

Relational and algebraic methods in computer science

This book constitutes the proceedings of the 14th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 2014 held in Marienstatt, Germany, in April/May 2014. The 25 revised full papers presented were carefully selected from 37 submissions. The papers are structured in specific fields on concurrent Kleene algebras and related formalisms, reasoning about computations and programs, heterogeneous and categorical approaches, applications of relational and algebraic methods and developments related to modal logics and lattices.

From Semirings to Residuated Kleene Lattices

We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-*. An investigation of congruence properties (e-permutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.

P 3 -isomorphisms for graphs

The P3-graph of a finite simple graph G is the graph whose vertices are the 3-vertex paths of G, with adjacency between two such paths whenever their union is a 4-vertex path or a 3-cycle. In this paper we show that connected fnite simple graphs G and H with isomorphic P3-graphs are either isomorphic or part of three exceptional families. We also characterize all isomorphisms between P3-graphs in terms of the original graphs.

Domain and Antidomain Semigroups

We axiomatise and study operations for relational domain and antidomain on semigroups and monoids. We relate this approach with previous axiomatisations for semirings, partial transformation semigroups and dynamic predicate logic.

Discriminator varieties of Boolean algebras with residuated operators

The theory of discriminator algebras and varieties has been investigated extensively, and provides us with a wealth of information and techniques applicable to specific examples of such algebras and varieties. Here we give several such examples for Boolean algebras with a residuated binary operator, abbreviated as r-algebras. More specifically, we show that all finite r-algebras, all integral ralgebras, all unital r-algebras with finitely many elements below the unit, and all commutative residuated monoids are discriminator algebras, provided they are subdirectly irreducible. These results are then used to give equational bases for some varieties of r-algebras. We also show that the variety of all residuated Boolean monoids is not a discriminator variety, which answers a question of B. Jonsson. 1. Preliminaries. A unary operation f on a Boolean algebra A0 = (A,+, 0, ·, 1,− ) is additive if f(x + y) = f(x) + f(y) and normal if f(0) = 0. For an n-ary operation f on A0, a sequence a ∈ A and i < n we define the (a, i)-translate of f to be the unary operation fa,i(x) = f(a0, . . . , ai−1, x, ai+1, . . . , an−1) . An operator on A0 is an n-ary operation for which all (a, i)-translates are additive and normal. Note that 0-ary operations (constants) have no translates, so they are operators by default. A = (A0,F) is a Boolean algebra with operators (BAO for short) if each f ∈ F is an operator on A0. The arity (or rank) of f is denoted by %f . To be an operator on a Boolean algebra is of course an equational property, and the variety of all BAOs with operators in F will be denoted by BAOF . The variety BAO{f}, where f is a unary operator, is usually referred to as the variety of modal algebras (the algebraic counterpart of modal logic). 1991 Mathematics Subject Classification: Primary 06E25; Secondary 03G15, 08A40, 03B45. The paper is in final form and no version of it will be published elsewhere.

Nonrepresentable Sequential Algebras

The sequential calculus of von Karger and Hoare [18] is designed for reasoning about sequential phenomena, dynamic or temporal logic, and concurrent or reactive systems. Unlike the classical calculus of relations, it has no operation for forming the converse of a relation. Sequential algebras [15] are algebras that satisfy certain equations in the sequential calculus. One standard example of a sequential algebra is the set of relations included in a partial ordering. Nonstandard examples arise by relativizing relation algebras to elements that are antisymmetric, transitive, and reflexive. The incompleteness and non-nite-axiomatizability of the sequential calculus are examined here from a relation-algebraic point of view. New constructions of nonrepresentable relation algebras are used to prove that there is no nite axiomatization of the equational theory of antisymmetric dense locally linear sequential algebras. The constructions improve on previous examples in certain interesting respects and give yet another proof that the classical calculus of relations is not nitely axiomatizable. 1