In abstract algebra, a partially ordered ring is a ring (A, +, ยท ), together with a compatible partial order, i.e. a partial order on the underlying set A that is compatible with the ring operations in the sense that it satisfies:
- implies
and
- and imply that
for all . Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring is a partially ordered ring where 's partially ordered additive group is Archimedean.
An ordered ring, also called a totally ordered ring, is a partially ordered ring where is additionally a total order.
An l-ring, or lattice-ordered ring, is a partially ordered ring where is additionally a lattice order.
Read more about Partially Ordered Ring: Properties, F-rings, Formally Verified Results For Commutative Ordered Rings
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