is a monoid, and
for each , the left and right multiplications by are residuated.
The second condition says: for every , each of the sets
is a down set, and has a maximum.
Clearly, and are both unique. is called the right residual of by , and is commonly denoted by , while is called the left residual of by , denoted by .
Residuated lattices are mostly found in algebraic structures associated with a variety of logical systems. For examples, Boolean algebras associated with classical propositional logic, and more generally Heyting algebras associated with the intuitionistic propositional logic are both residuated, with multiplication the same as the lattice meet operation. MV-algebras and BL-algebras associated with many-valued logics are further examples of residuated lattices.
Remark. A residuated lattice is said to be commutative 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 Logic: Semantic, Algebras, and Derivation Systems, Cambridge University Press (2008)
- 3 R. P. Dilworth, M. Ward Residuated Lattices, Transaction of the American Mathematical Society 45, pp.335-354 (1939)
|Date of creation||2013-03-22 18:53:41|
|Last modified on||2013-03-22 18:53:41|
|Last modified by||CWoo (3771)|
|Defines||commutative residuated lattice|