Abbreviation: ORng

An \emph{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 \emph{order-preserving}: $x\le y\Longrightarrow x+z\le y+z$

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

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

An \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.

None

[[Complete ordered rings]]

[[Ordered fields]]

[[Abelian ordered groups]] reduced type

[[Ordered monoids]] reduced type

[[Rings]] reduced type