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Ordered Algebraic Structures pp 19–56Cite as

A Survey of Residuated Lattices

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Part of the book series: Developments in Mathematics ((DEVM,volume 7))

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.

We would like to thank Jac Cole, Nick Galatos, Tomasz Kowalski, Hiroakira Ono, James Raftery and an anonymous referee for numerous observations and suggestions that have substantially improved this survey.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    P. Jipsen & C. Tsinakis

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  1. P. Jipsen
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  2. C. Tsinakis
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Editors and Affiliations

  1. Department of Mathematics, University of Florida, Gainesville, Florida, USA

    Jorge Martínez

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Jipsen, P., Tsinakis, C. (2002). A Survey of Residuated Lattices. In: Martínez, J. (eds) Ordered Algebraic Structures. Developments in Mathematics, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3627-4_3

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  • DOI: https://doi.org/10.1007/978-1-4757-3627-4_3

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-5225-7

  • Online ISBN: 978-1-4757-3627-4

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