Chapter

Ordered Algebraic Structures

Volume 7 of the series Developments in Mathematics pp 19-56

A Survey of Residuated Lattices

  • P. JipsenAffiliated withDepartment of Mathematics, Vanderbilt University
  • , C. TsinakisAffiliated withDepartment of Mathematics, Vanderbilt University

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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.