## 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}: $xy=yx$

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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 …: $axiom$

Example 1:

### 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.

Classtype (value, see description) 1) yes yes yes, $n=2$ yes yes

### Subclasses

[[Commutative f-rings]] subvariety

### Superclasses

[[Lattice-ordered rings]] supervariety
[[Abelian lattice-ordered groups]] subreduct
[[Commutative rings]] subreduct

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