Abbreviation: CanRL

A \emph{cancellative residuated lattice} is a residuated lattice $\mathbf{L}=\langle L, \vee, \wedge, \cdot, e, \backslash, /\rangle$ such that

$\cdot$ is right-cancellative: $xz=yz\Longrightarrow x=y$

$\cdot$ is left-cancellative: $zx=zy\Longrightarrow x=y$

Let $\mathbf{L}$ and $\mathbf{M}$ be cancellative residuated lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ $h(x\cdot y)=h(x)\cdot h(y)$, $h(x\backslash y)=h(x)\backslash h(y)$, $h(x/y)=h(x)/h(y)$ and $h(e)=e$

Example 1:

$\begin{array}{lr} None \end{array}$

Cancellative commutative residuated lattices

Cancellative distributive residuated lattices

Residuated lattices