Ordered rings

Abbreviation: ORng

Definition

An ordered ring is a structure $\mathbf{A}=\langle A,+,-,0,\cdot,1,\le\rangle$ such that

$\langle A,+,-,0,\cdot,1\rangle$ is a ring

$\langle A,\le\rangle$ is a linear order

$+$ is order-preserving: $x\le y\Longrightarrow x+z\le y+z$

$\cdot$ is order-preserving for positive elements: $x\le y\text{ and }0\le z\Longrightarrow xz\le yz$

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 … . A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x ... y)=h(x) ... h(y)$

Definition

An 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

None

Subclasses

[[Complete ordered rings]]
[[Ordered fields]]

Superclasses

[[Abelian ordered groups]] reduced type
[[Ordered monoids]] reduced type
[[Rings]] reduced type

References