residuated lattice
A residuated lattice is a lattice^{} $L$ with an additional binary operation^{} $\cdot $ called multiplication^{}, with a multiplicative identity^{} $e\in L$, such that

•
$(L,\cdot ,e)$ is a monoid, and

•
for each $x\in L$, the left and right multiplications by $x$ are residuated.
The second condition says: for every $x,z\in L$, each of the sets
$$L(x,z):=\{y\in L\mid x\cdot y\le z\}$$ 
and
$$R(x,z):=\{y\in L\mid y\cdot x\le z\}$$ 
is a down set, and has a maximum.
Clearly, $\mathrm{max}L(x,z)$ and $\mathrm{max}R(x,z)$ are both unique. $\mathrm{max}L(x,z)$ is called the right residual of $z$ by $x$, and is commonly denoted by $x\backslash z$, while $\mathrm{max}R(x,z)$ is called the left residual of $z$ by $x$, denoted by $x/z$.
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^{}. MValgebras and BLalgebras associated with manyvalued logics are further examples of residuated lattices.
Remark. A residuated lattice is said to be commutative^{} if $\cdot $ 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 ManyValued 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.335354 (1939)
Title  residuated lattice 

Canonical name  ResiduatedLattice 
Date of creation  20130322 18:53:41 
Last modified on  20130322 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 