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: $x\le y\text{ and }0\le z\Longrightarrow xz\le yz$

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 …: $axiom$

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 universal

None

Subclasses

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

Superclasses

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