Partially ordered ring (Topic)

In abstract algebra, a partially ordered ring is a ring (A, +, ·), 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: $${\\displaystyle x\\leq y{\\text{ implies }}x+z\\leq y+z}$$ and $${\\displaystyle 0\\leq x{\\text{ and }}0\\leq y{\\text{ imply that }}0\\leq x\\cdot y}$$ for all ${\\displaystyle x,y,z\\in A}$. 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 ${\\displaystyle (A,\\leq )}$ where partially ordered additive group is Archimedean.