On Birkhoff's common abstraction problem

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a “common abstraction” that includes Boolean algebras and latticeordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA∨LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA∨LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA ∨ LG

Cancellative residuated lattices as lattice ordered groups with a modality

Our work proposes a new paradigm for the study of various classes of cancellative residuated lattices by viewing these structures as lattice-ordered groups with a suitable operator (a conucleus). One consequence of our approach is the categorical equivalence between the variety of cancellative commutative residuated lattices and the category of abelian lattice-ordered groups endowed with a conucleus whose image generates the underlying group of the lattice-ordered group. In addition, we extend our methods to obtain a categorical equivalence between PiMTL-algebras and product algebras with a conucleus. Among the other results of the paper, we single out the introduction of a categorical framework for making precise the view that some of the most interesting algebras arising in algebraic logic are related to lattice-ordered groups. More specifically, we show that these algebras are subobjects and quotients of lattice-ordered groups in a ``quantale like'' category of algebras

On Birkhoff’s “common abstraction” problem

In his milestone textbook Lattice Theory, Garrett Birkhoff challenged his readers to develop a "common abstraction" that includes Boolean algebras and lattice-ordered groups as special cases. In this paper, after reviewing the past attempts to solve the problem, we provide our own answer by selecting as common generalization of BA and LG their join BA∨LG in the lattice of subvarieties of FL (the variety of FL-algebras); we argue that such a solution is optimal under several respects and we give an explicit equational basis for BA ∨ LG relative to FL. Finally, we prove a Holland-type representation theorem for a variety of FL-algebras containing BA ∨ LG

The structure of residuated lattices

Abstract. 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. We end with a list of open problems that we hope will stimulate further research.