nLab strictly weakly ordered ring

Context

Algebra

(0,1)-Category theory

Contents

Idea

A notion of ordered ring for strict weak orders.

Definition

With an order relation

A strictly weakly ordered ring is an ring RR with a strict weak order <\lt such that

  • 0<10 \lt 1

  • for all aRa \in R and bRb \in R, 0<a0 \lt a and 0<b0 \lt b implies that 0<a+b0 \lt a + b; alternatively, 0<a+b0 \lt a + b implies that 0<a0 \lt a or 0<b0 \lt b.

  • for all aRa \in R and bRb \in R, if 0<a0 \lt a and 0<b0 \lt b, then 0<ab0 \lt a \cdot b

We define the predicate isPositive\mathrm{isPositive} as

isPositive(a)0<a\mathrm{isPositive}(a) \coloneqq 0 \lt a

With a positivity predicate

Let RR be a commutative ring. RR is an pseudo-ordered ring if there is a predicate isPositive\mathrm{isPositive} stating an element aa is positive, such that

  • 0 is not positive
  • given element aRa \in R, if aa is positive, then a-a is not positive
  • given elements aRa \in R and bRb \in R; if aa is positive, then either bb is positive or aba - b is positive
  • 1 is positive
  • given elements aRa \in R and bRb \in R, if aa is positive and bb is positive, a+ba + b is positive; alternatively, a+ba + b being positive implies that aa is positive or bb is positive.
  • given elements aRa \in R and bRb \in R, if aa is positive and bb is positive, aba \cdot b is positive

We define the order relation <\lt as

a<bisPositive(ba)a \lt b \coloneqq \mathrm{isPositive}(b - a)

Properties

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.

Examples

See also

Last revised on August 19, 2024 at 15:13:42. See the history of this page for a list of all contributions to it.