Abbreviation: CLRng

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.

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: $h(x \vee y)=h(x) \vee h(y)$, $h(x \wedge y)=h(x) \wedge h(y)$, $h(x + y)=h(x) + h(y)$, $h(x \cdot y)=h(x) \cdot h(y)$.

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:

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.

[[Commutative f-rings]] subvariety

[[Lattice-ordered rings]] supervariety

[[Abelian lattice-ordered groups]] subreduct

[[Commutative rings]] subreduct