### Table of Contents

## Commutative lattice-ordered rings

Abbreviation: **CLRng**

### Definition

A \emph{commutative lattice-ordered ring} is a lattice-ordered ring $\mathbf{A}=\langle A,\vee,\wedge,+,-,0,\cdot\rangle$ such that

$\cdot$ is \emph{commutative}: $xy=yx$

Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'' link at the bottom and fill out this page.

It is not unusual to give several (equivalent) definitions. Ideally, one of the definitions would give an irredundant axiomatization that does not refer to other classes.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be commutative lattice-ordered rings. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x + y)=h(x) + h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$.

### Definition

A \emph{…} is a structure $\mathbf{A}=\langle A,\ldots\rangle$ of type $\langle …\rangle$ such that

$\ldots$ is …: $axiom$

$\ldots$ is …: $axiom$

### Examples

Example 1:

### Basic results

### Properties

Feel free to add or delete properties from this list. The list below may contain properties that are not relevant to the class that is being described.

### Finite members

### Subclasses

[[Commutative f-rings]] subvariety

### Superclasses

[[Lattice-ordered rings]] supervariety

[[Abelian lattice-ordered groups]] subreduct

[[Commutative rings]] subreduct