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 on the underlying set A that is compatible with the ring operations in the sense that it satisfies: and 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. The additive group of a partially ordered ring is always a partially ordered group. The set of non-negative elements of a partially ordered ring (the set of elements for which also called the positive cone of the ring) is closed under addition and multiplication, that is, if is the set of non-negative elements of a partially ordered ring, then and Furthermore, The mapping of the compatible partial order on a ring 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 is a subset of a ring and: then the relation where if and only if defines a compatible partial order on (that is, is a partially ordered ring). In any l-ring, the of an element can be defined to be where denotes the maximal element. For any and holds. An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring in which and imply that for all 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.