symmetric monoidal (∞,1)-category of spectra
A notion of ordered ring for pseudo-orders/strict total orders.
A pseudo-ordered ring or a strictly totally ordered ring is an ring $R$ with a pseudo-order/strict total 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 $0 \lt a \cdot b$
Every pseudo-ordered ring is a partially ordered ring given by the negation of the strict total order. In the presence of excluded middle, every pseudo-ordered ring is a totally ordered ring.
Pseudo-ordered rings may have zero divisors. The pseudo-ordered rings which do not have zero divisors are ordered integral domains.
the integers are a pseudo-ordered ring
the Dedekind real numbers are a pseudo-ordered ring which in constructive mathematics cannot be proved to be a totally ordered ring.
Last revised on February 1, 2024 at 19:00:05. See the history of this page for a list of all contributions to it.