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].

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

For the entire collection see [Zbl 1068.06001].

##### MSC:

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 |