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, $\max L(x,z)$ 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 $x\backslash \backslash z$, 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.

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