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Partially ordered ring

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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 .[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

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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;[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 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).[2]

In any l-ring, the absolute value of an element can be defined to be where denotes the maximal element. For any and holds.[3]

f-rings

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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 not positive, even though it is a square.[2] The additional hypothesis required of f-rings eliminates this possibility.

Example

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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: [2]

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

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  • 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 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.[2] Some mathematicians take this to be the definition of an f-ring.[3]

Formally verified results for commutative ordered rings

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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 is an ordered group OrdRing_ZF_1_L4
OrdRing_ZF_1_L7
and imply
and
OrdRing_ZF_1_L9
ordring_one_is_nonneg
OrdRing_ZF_2_L5
ord_ring_triangle_ineq
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 if and only if has no zero divisors. OrdRing_ZF_3_L3
If is non-trivial (), then it is infinite. ord_ring_infinite

See also

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References

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  1. ^ a b c Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7.
  2. ^ a b c d e f 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.
  3. ^ a b c d 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.
  4. ^ denotes infimum.
  5. ^ 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.
  6. ^ "IsarMathLib" (PDF). Retrieved 2009-03-31.

Further reading

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  • 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
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