A notion of ordered ring for strict weak orders.
A strictly weakly ordered ring is an ring $R$ with a strict weak order $\lt$ such that
$0 \lt 1$
for all $a \in R$ and $b \in R$, $0 \lt a$ and $0 \lt b$ implies that $0 \lt a + b$; alternatively, $0 \lt a + b$ implies that $0 \lt a$ or $0 \lt b$.
for all $a \in R$ and $b \in R$, if $0 \lt a$ and $0 \lt b$, then $0 \lt a \cdot b$
We define the predicate $\mathrm{isPositive}$ as
Let $R$ be a commutative ring. $R$ is an pseudo-ordered ring if there is a predicate $\mathrm{isPositive}$ stating an element $a$ is positive, such that
We define the order relation $\lt$ as
Every strictly weakly ordered ring is a preordered ring given by the negation of the strict weak order. In the presence of excluded middle, every strictly weakly ordered ring is a totally preordered ring.
Every pseudo-ordered ring is a strictly weakly ordered ring.
Every ordered local ring is a strictly weakly ordered ring where every element greater than zero or less than zero is invertible.
In particular, the set of Cauchy sequences of rational numbers is a strictly weakly ordered ring.
Every ordered Kock field is a example of a strictly weakly ordered ring which in the presence of excluded middle is a pseudo-ordered ring.
Last revised on August 19, 2024 at 15:13:42. See the history of this page for a list of all contributions to it.