=====Cancellative residuated lattices=====

Abbreviation: **CanRL**
====Definition====
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:  $zx=zy\Longrightarrow x=y$

==Morphisms==
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 $h(e)=e$

====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |no |
^[[Congruence e-regular]]  |yes |
^[[Congruence uniform]]  |no |
^[[Congruence extension property]]  |no |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |

====Finite members====

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


====Subclasses====
[[Cancellative commutative residuated lattices]] 

[[Cancellative distributive residuated lattices]] 

====Superclasses====
[[Residuated lattices]] 


====References====

[(Ln19xx>
)]