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In abstract algebra, a partially ordered ring is a ring , together with a compatible partial order, that is, a partial order ≤ {\displaystyle \,\leq \,} on the underlying set A that is compatible with the ring operations in the sense that it satisfies:
An ordered ring, also called a totally ordered ring, is a partially ordered ring {\displaystyle } where ≤ {\displaystyle \,\leq \,} is additionally a total order.
An l-ring, or lattice-ordered ring, is a partially ordered ring {\displaystyle } where ≤ {\displaystyle \,\leq \,} is additionally a lattice order.
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