Residuated lattice

Remarks

Clause 3 says that $x \backslash z$ is the largest $y$ with $x \cdot y \le z$, and $z / y$ is the largest $x$ with $x \cdot y \le z$. Example: a Heyting algebra $H$ with $x \cdot y = x \wedge y$, $e = 1$ and $x \backslash z = z / x = x \to z$ is a residuated lattice. Clause 3 is then the defining rule $x \wedge y \le z \iff y \le x \to z$ of the Heyting implication. Example: the group $\Z$ with $\min$, $\max$, $x \cdot y = x + y$, $e = 0$ and $x \backslash z = z / x = z - x$ is a residuated lattice, since $x + y \le z \iff y \le z - x$. Non-example: $\N$ with $\min$, $\max$, ordinary multiplication and $e = 1$ satisfies clauses 1 and 2, but has no left residual. For $x = 0$ and $z = 0$, every $y \in \N$ satisfies $0 \cdot y \le 0$, and $\N$ has no largest element. Many authors also require $L$ to be bounded, or the monoid to be commutative, in which case $x \backslash z = z / x$ is written $x \to z$. Multiplication in a residuated lattice is monotone and distributes over all existing joins.