Commutative lattice-ordered rings

Abbreviation: CLRng

Definition

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

$\cdot$ is 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 is a structure $\mathbf{A}=\langle A,...\rangle$ of type $\langle ...\rangle$ such that

$...$ is …: $axiom$

$...$ is …: $axiom$

Examples

Example 1:

Basic results

Properties

Finite members

Subclasses

[[Commutative f-rings]] subvariety

Superclasses

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

References