A notion of ordered ring for strict weak orders.
A strictly weakly ordered ring is an ring with a strict weak order such that
for all and , and implies that ; alternatively, implies that or .
for all and , if and , then
We define the predicate as
Let be a commutative ring. is an pseudo-ordered ring if there is a predicate stating an element is positive, such that
We define the order relation 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.