↑ arvau

Wikipedia / 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. Various extensions of this definition exist that constrain the ring, the partial order, or both. For example, an Archimedean partially ordered ring where partially ordered additive group is Archimedean.

An ordered ring, also called a totally ordered ring is additionally a total order.

An l-ring, or lattice-ordered ring is additionally a lattice order.

Properties

The additive group of a partially ordered ring is always a partially ordered group.

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.

is a subset and:

if and only if is a partially ordered ring).

In any l-ring, the denotes the maximal element holds.

f-rings

An f-ring, or PierceBirkhoff ring 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

be a Hausdorff space be the space of all continuous, real-valued functions

are fairly rigid. For example, localisations 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.

See also

References

Further reading