partially ordered ring

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order \,\leq\, on the underlying set A that is compatible with the ring operations in the sense that it satisfies: x \leq y \text{ implies } x + z \leq y + z and 0 \leq x \text{ and } 0 \leq y \text{ imply that } 0 \leq x \cdot y for all 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 (A, \leq) where partially ordered additive group is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a total order. An l-ring, or lattice-ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a lattice order.

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order \,\leq\, on the underlying set A that is compatible with the ring operations in the sense that it satisfies: x \leq y \text{ implies } x + z \leq y + z and 0 \leq x \text{ and } 0 \leq y \text{ imply that } 0 \leq x \cdot y for all 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 (A, \leq) where partially ordered additive group is Archimedean. An ordered ring, also called a totally ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a total order. An l-ring, or lattice-ordered ring, is a partially ordered ring (A, \leq) where \,\leq\, is additionally a lattice order. The set of non-negative elements of a partially ordered ring (the set of elements x for which 0 \leq x, also called the positive cone of the ring) is closed under addition and multiplication, that is, if P is the set of non-negative elements of a partially ordered ring, then P + P \subseteq P and P \cdot P \subseteq P. Furthermore, P \cap (-P) = \{0\}. The mapping of the compatible partial order on a ring A to the set of its non-negative elements is one-to-one; that is, the compatible partial order uniquely determines the set of non-negative elements, and a set of elements uniquely determines the compatible partial order if one exists. If S \subseteq A is a subset of a ring A, and: 0 \in S S \cap (-S) = \{0\} S + S \subseteq S S \cdot S \subseteq S then the relation \,\leq\, where x \leq y if and only if y - x \in S defines a compatible partial order on A (that is, (A, \leq) is a partially ordered ring). In any l-ring, the |x| of an element x can be defined to be x \vee(-x), where x \vee y denotes the maximal element. For any x and y, |x \cdot y| \leq |x| \cdot |y| holds. An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring (A, \leq) in which x \wedge y = 0 and 0 \leq z imply that zx \wedge y = xz \wedge y = 0 for all x, y, z \in A. They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings", in an attempt to restrict the class of l-rings so as to eliminate a number of pathological examples. For example, Birkhoff and Pierce demonstrated an l-ring with 1 in which 1 is not positive, even though it is a square. The additional hypothesis required of f-rings eliminates this possibility. [f + g](x) = f(x) + g(x) [fg](x) = f(x) \cdot g(x) [f \wedge g](x) = f(x) \wedge g(x). From an algebraic point of view the rings \mathcal{C}(X) are fairly rigid. For example, localisations, residue rings or limits of rings of the form \mathcal{C}(X) are not of this form in general. A much more flexible class of f-rings containing all rings of continuous functions and resembling many of the properties of these rings is the class of real closed rings. |xy| = |x||y| in an f-ring. The category Arf consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity. Every ordered ring is an f-ring, so every sub-direct union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a sub-direct union of ordered rings. Some mathematicians take this to be the definition of an f-ring.

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