- algebra, higher algebra
- universal algebra
- monoid, semigroup, quasigroup
- nonassociative algebra
- associative unital algebra
- commutative algebra
- Lie algebra, Jordan algebra
- Leibniz algebra, pre-Lie algebra
- Poisson algebra, Frobenius algebra
- lattice, frame, quantale
- Boolean ring, Heyting algebra
- commutator, center
- monad, comonad
- distributive law

- group, normal subgroup
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A totally ordered ring is an ordered ring whose order forms a total order.

This definition is adapted from Peter Freyd‘s definition of a totally ordered abelian group:

A **totally ordered ring** is an lattice-ordered ring $R$ such that for all elements $a$ in $R$, $a \leq 0$ or $-a \leq 0$.

In a totally ordered ring, the join is usually called the **maximum**, while the meet is usually called the **minimum**

If the relation $\leq$ is only a preorder, then the prelattice-ordered ring $R$ is said to be a **totally preordered ring**.

The integers, the rational numbers, and the real numbers are totally ordered rings.

- Peter Freyd,
*Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 (tac:20-10)

- Wikipedia,
*Ordered ring*

Last revised on August 19, 2024 at 15:13:55. See the history of this page for a list of all contributions to it.