residuated lattice
A residuated lattice is a lattice with an additional binary operation
called multiplication
, with a multiplicative identity
, such that
-
•
is a monoid, and
-
•
for each , the left and right multiplications by are residuated.
The second condition says: for every , each of the sets
and
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.
References
- 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)
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 |