Partially ordered ring
In abstract algebra, a partially ordered ring is a ring, 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 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.
Properties
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 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:
In any l-ring, the of an element can be defined to be where denotes the maximal element. For any and
holds.
f-rings
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. The additional hypothesis required of f-rings eliminates this possibility.Example
Let be a Hausdorff space, and be the space of all continuous, real-valued functions on is an Archimedean f-ring with 1 under the following pointwise operations:From an algebraic point of view the rings
are fairly rigid. For example, localisations, residue rings or limits of rings of the form 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.
Properties
- A direct product of f-rings is an f-ring, an l-subring of an f-ring is an f-ring, and an l-homomorphic image of an f-ring is an f-ring.
- 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.