residuated lattice

A residuated lattice is a latticeMathworldPlanetmathPlanetmath L with an additional binary operationMathworldPlanetmath called multiplicationPlanetmathPlanetmath, with a multiplicative identityPlanetmathPlanetmath eL, such that

  • (L,,e) is a monoid, and

  • for each xL, the left and right multiplications by x are residuated.

The second condition says: for every x,zL, each of the sets




is a down set, and has a maximum.

Clearly, maxL(x,z) and maxR(x,z) are both unique. maxL(x,z) is called the right residual of z by x, and is commonly denoted by x\z, while maxR(x,z) is called the left residual of z by x, denoted by x/z.

Residuated lattices are mostly found in algebraic structuresPlanetmathPlanetmath associated with a varietyMathworldPlanetmathPlanetmath of logical systems. For examples, Boolean algebrasMathworldPlanetmath associated with classical propositional logicPlanetmathPlanetmath, and more generally Heyting algebrasMathworldPlanetmath associated with the intuitionistic propositional logic are both residuated, with multiplication the same as the lattice meet operationMathworldPlanetmath. MV-algebras and BL-algebras associated with many-valued logics are further examples of residuated lattices.

Remark. A residuated lattice is said to be commutativePlanetmathPlanetmathPlanetmath if is commutative. All of the examples cited above are commutative.


  • 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005)
  • 2 M. Bergmann, An Introduction to Many-Valued and Fuzzy LogicMathworldPlanetmath: Semantic, AlgebrasMathworldPlanetmathPlanetmath, and DerivationPlanetmathPlanetmath Systems, Cambridge University Press (2008)
  • 3 R. P. Dilworth, M. Ward Residuated Lattices, Transaction of the American Mathematical Society 45, pp.335-354 (1939)
Title residuated lattice
Canonical name ResiduatedLattice
Date of creation 2013-03-22 18:53:41
Last modified on 2013-03-22 18:53:41
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 9
Author CWoo (3771)
Entry type Definition
Classification msc 06B99
Defines left residual
Defines right residual
Defines commutative residuated lattice