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#### Abbreviation: CanRL

### Definition

A*\ cancellative residuated lattice* is a residuated lattice *L* = (*L*,∨,∧,·,*e*,**\**,**/**) such that

· is right-cancellative: *x**z* = *y**z* ⇒ *x* = *y*, and

· is left-cancellative: *z**x* = *z**y* ⇒ *x* = *y*.

**Remark**:

### Morphisms

Let *L* and *M* be cancellative residuated lattices. A
morphism from *L* to *M* is a function *h* : *L*→*M*
that is a homomorphism:
*h*(*x*∨*y*) = *h*(*x*)∨*h*(*y*) and *h*(*x*∧*y*) = *h*(*x*)∧*h*(*y*) and *h*(*x*·*y*) = *h*(*x*)·*h*(*y*) and *h*(*x***\***y*) = *h*(*x*)**\***h*(*y*) and *h*(*x***/***y*) = *h*(*x*)**/***h*(*y*) and *h*(*e*) = *e*.

### Some results

### Examples

### Properties

### Finite members

None

### Subclasses

[Cancellative commutative residuated lattices]?

[Cancellative distributive residuated lattices]?

### Superclasses

Residuated lattices