Partially ordered ring – Wikipedia

Ring with a compatible partial order

In abstract algebra, a partially ordered ring is a ring (A, +, ·), together with a compatible partial order, that is, a partial order

${displaystyle ,leq ,}$

on the underlying set A that is compatible with the ring operations in the sense that it satisfies:

and

for all

${displaystyle x,y,zin A}$

.^[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

${displaystyle (A,leq )}$

where

${displaystyle A}$

‘s partially ordered additive group is Archimedean.^[2]

An ordered ring, also called a totally ordered ring, is a partially ordered ring

${displaystyle (A,leq )}$

where

${displaystyle ,leq ,}$

is additionally a total order.^[1][2]

An l-ring, or lattice-ordered ring, is a partially ordered ring

${displaystyle (A,leq )}$

where

${displaystyle ,leq ,}$

is additionally a lattice order.

Properties[edit]

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

${displaystyle x}$

for which

${displaystyle 0leq x,}$

also called the positive cone of the ring) is closed under addition and multiplication, that is, if

${displaystyle P}$

is the set of non-negative elements of a partially ordered ring, then

${displaystyle P+Psubseteq P}$

and

${displaystyle Pcdot Psubseteq P.}$

Furthermore,

${displaystyle Pcap (-P)={0}.}$

The mapping of the compatible partial order on a ring

${displaystyle A}$

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

${displaystyle Ssubseteq A}$

is a subset of a ring

${displaystyle A,}$

and:

1. 
   ${displaystyle 0in S}$

   

2. 
   ${displaystyle Scap (-S)={0}}$

   

3. 
   ${displaystyle S+Ssubseteq S}$

   

4. 
   ${displaystyle Scdot Ssubseteq S}$

   

then the relation

${displaystyle ,leq ,}$

where

${displaystyle xleq y}$

if and only if

${displaystyle y-xin S}$

defines a compatible partial order on

${displaystyle A}$

(that is,

${displaystyle (A,leq )}$

is a partially ordered ring).^[2]

In any l-ring, the absolute value

${displaystyle |x|}$

of an element

${displaystyle x}$

can be defined to be

${displaystyle xvee (-x),}$

where

${displaystyle xvee y}$

denotes the maximal element. For any

${displaystyle x}$

and

${displaystyle y,}$

holds.^[3]

f-rings[edit]

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring

${displaystyle (A,leq )}$

in which

${displaystyle xwedge y=0}$

^[4] and

${displaystyle 0leq z}$

imply that

${displaystyle zxwedge y=xzwedge y=0}$

for all

${displaystyle x,y,zin A.}$

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[edit]

Let

${displaystyle X}$

be a Hausdorff space, and

${displaystyle {mathcal {C}}(X)}$

be the space of all continuous, real-valued functions on

${displaystyle X.}$

${displaystyle {mathcal {C}}(X)}$

is an Archimedean f-ring with 1 under the following pointwise operations:

^[2]

From an algebraic point of view the rings

${displaystyle {mathcal {C}}(X)}$

are fairly rigid. For example, localisations, residue rings or limits of rings of the form

${displaystyle {mathcal {C}}(X)}$

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[edit]

• 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]

• 
  ${displaystyle |xy|=|x||y|}$

  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[edit]

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

${displaystyle (A,leq )}$

is a commutative ordered ring, and

${displaystyle x,y,zin A.}$

Then:

See also[edit]

References[edit]

Further reading[edit]

• 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[edit]