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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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
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