# residuated lattice

• $(L,\cdot,e)$ 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

 $L(x,z):=\{y\in L\mid x\cdot y\leq z\}$

and

 $R(x,z):=\{y\in L\mid y\cdot x\leq z\}$

is a down set, and has a maximum.

Clearly, $\max L(x,z)$ and $\max R(x,z)$ are both unique. $\max L(x,z)$ is called the right residual of $z$ by $x$, and is commonly denoted by $x\backslash z$, while $\max R(x,z)$ 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  . MV-algebras and BL-algebras associated with many-valued logics are further examples of residuated lattices.

## References

• 1 T.S. Blyth, Lattices and Ordered Algebraic Structures, Springer, New York (2005)
• 2
• 3 R. P. Dilworth, M. Ward Residuated Lattices, Transaction of the American Mathematical Society 45, pp.335-354 (1939)
Title residuated lattice ResiduatedLattice 2013-03-22 18:53:41 2013-03-22 18:53:41 CWoo (3771) CWoo (3771) 9 CWoo (3771) Definition msc 06B99 left residual right residual commutative residuated lattice