A survey of residuated lattices. (English) Zbl 1070.06005

MartĂ­nez, Jorge (ed.), Ordered algebraic structures. Proceedings of the conference on lattice-ordered groups and \(f\)-rings held at the University of Florida, Gainesville, FL, USA, February 28–March 3, 2001. Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-0752-3). Developments in Mathematics 7, 19-56 (2002).
Summary: 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.
For the entire collection see [Zbl 1068.06001].


06F05 Ordered semigroups and monoids
03B25 Decidability of theories and sets of sentences
08B15 Lattices of varieties
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures