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:
- and imply that
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 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; 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).
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,
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 negative, even though being a square. The additional hypothesis required of f-rings eliminates this possibility.
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.
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 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. Some mathematicians take this to be the definition of an f-ring.
Formally verified results for commutative ordered rings
Suppose is a commutative ordered ring, and . Then:
|The additive group of A is an ordered group||OrdRing_ZF_1_L4|
| and imply
|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|
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- 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.
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- 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