24 research outputs found
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.