an:02208113
Zbl 1070.06005
Jipsen, P.; Tsinakis, C.
A survey of residuated lattices
EN
Mart??nez, 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).
2002
a
06F05 03B25 08B15 06-02
survey; residuated monoid operation; residuated lattices; lattice of varieties; decidability
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].