# Partially ordered ring

In abstract algebra, a **partially ordered ring** is a ring (*A*, +, **·** ), together with a *compatible partial order*, i.e. a partial order on the underlying set *A* that is compatible with the ring operations in the sense that it satisfies:

- implies

and

- and imply that

for all .[1] 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.[2]

An **ordered ring**, also called a **totally ordered ring**, is a partially ordered ring where is additionally a total order.[1][2]

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 (the set of elements *x* for which , also called the positive cone of the ring) is closed under addition and multiplication, i.e., if *P* 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 *A* to the set of its non-negative elements is one-to-one;[1] 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* is a subset of a ring *A*, and:

then the relation where iff defines a compatible partial order on *A* (*ie.* is a partially ordered ring).[2]

In any l-ring, the *absolute value* of an element *x* can be defined to be , where denotes the maximal element. For any *x* and *y*,

holds.[3]

## f-rings

An **f-ring**, or **Pierce–Birkhoff ring**, is a lattice-ordered ring in which [4] 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 negative, even though being a square.[2] The additional hypothesis required of f-rings eliminates this possibility.

### Example

Let *X* be a Hausdorff space, and be the space of all continuous, real-valued functions on *X*. is an Archimedean f-ring with 1 under the following point-wise 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.[3]

in an f-ring.[3]

The category **Arf** consists of the Archimedean f-rings with 1 and the l-homomorphisms that preserve the identity.[5]

Every ordered ring is an f-ring, so every subdirect 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 subdirect union of ordered rings.[2] Some mathematicians take this to be the definition of an f-ring.[3]

## Formally verified results for commutative ordered rings

IsarMathLib, a library for the Isabelle theorem prover, has formal verifications of a few fundamental results on commutative ordered rings. The results are proved in the `ring1` context.[6]

Suppose is a commutative ordered ring, and . Then:

by | |
---|---|

The additive group of A is an ordered group |
OrdRing_ZF_1_L4 |

iff | OrdRing_ZF_1_L7 |

and imply and |
OrdRing_ZF_1_L9 |

ordring_one_is_nonneg | |

OrdRing_ZF_2_L5 | |

ord_ring_triangle_ineq | |

x is either in the positive set, equal to 0, or in minus the positive set. |
OrdRing_ZF_3_L2 |

The set of positive elements of is closed under multiplication iff A has no zero divisors. |
OrdRing_ZF_3_L3 |

If A is non-trivial (), then it is infinite. |
ord_ring_infinite |

## References

- Anderson, F. W. "Lattice-ordered rings of quotients".
*Canadian Journal of Mathematics*.**17**: 434–448. doi:10.4153/cjm-1965-044-7. - Johnson, D. G. (December 1960). "A structure theory for a class of lattice-ordered rings".
*Acta Mathematica*.**104**(3–4): 163–215. doi:10.1007/BF02546389. - Henriksen, Melvin (1997). "A survey of f-rings and some of their generalizations". In W. Charles Holland and Jorge Martinez (ed.).
*Ordered Algebraic Structures: Proceedings of the Curaçao Conference Sponsored by the Caribbean Mathematics Foundation, June 23–30, 1995*. the Netherlands: Kluwer Academic Publishers. pp. 1–26. ISBN 0-7923-4377-8. - denotes infimum.
- Hager, Anthony W.; Jorge Martinez (2002). "Functorial rings of quotients—III: The maximum in Archimedean f-rings".
*Journal of Pure and Applied Algebra*.**169**: 51–69. doi:10.1016/S0022-4049(01)00060-3. - "IsarMathLib" (PDF). Retrieved 2009-03-31.

## Further reading

- Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings".
*Anais da Academia Brasileira de Ciências*.**28**: 41–69. - Gillman, Leonard; Jerison, Meyer Rings of continuous functions. Reprint of the 1960 edition. Graduate Texts in Mathematics, No. 43. Springer-Verlag, New York-Heidelberg, 1976. xiii+300 pp

## External links

- "Ordered Ring, Partially Ordered Ring". Encyclopedia of Mathematics. Retrieved 2009-04-03.
- "Partially Ordered Ring". PlanetMath. Retrieved 2018-04-14.