Partially Ordered Ring

In abstract algebra, a partially ordered ring is a ring (A, +, · ), 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.