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
x,y,z∈A{displaystyle x,y,zin A}(A,≤){displaystyle (A,leq )} .[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
A{displaystyle A} where
‘s partially ordered additive group is Archimedean.[2]
An ordered ring, also called a totally ordered ring, is a partially ordered ring
(A,≤){displaystyle (A,leq )}≤{displaystyle ,leq ,} where
is additionally a total order.[1][2]
An l-ring, or lattice-ordered ring, is a partially ordered ring
(A,≤){displaystyle (A,leq )}≤{displaystyle ,leq ,} where
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
x{displaystyle x}0≤x,{displaystyle 0leq x,} for which
P{displaystyle P} also called the positive cone of the ring) is closed under addition and multiplication, that is, if
P+P⊆P{displaystyle P+Psubseteq P} is the set of non-negative elements of a partially ordered ring, then
P⋅P⊆P.{displaystyle Pcdot Psubseteq P.} and
P∩(−P)={0}.{displaystyle Pcap (-P)={0}.} Furthermore,
The mapping of the compatible partial order on a ring
A{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
S⊆A{displaystyle Ssubseteq A}A,{displaystyle A,} is a subset of a ring
and:
0∈S{displaystyle 0in S}
S∩(−S)={0}{displaystyle Scap (-S)={0}}
S+S⊆S{displaystyle S+Ssubseteq S}
S⋅S⊆S{displaystyle Scdot Ssubseteq S}
then the relation
≤{displaystyle ,leq ,}x≤y{displaystyle xleq y} where
y−x∈S{displaystyle y-xin S} if and only if
A{displaystyle A} defines a compatible partial order on
(A,≤){displaystyle (A,leq )} (that is,
is a partially ordered ring).[2]
In any l-ring, the absolute value
|x|{displaystyle |x|}x{displaystyle x} of an element
x∨(−x),{displaystyle xvee (-x),} can be defined to be
x∨y{displaystyle xvee y} where
x{displaystyle x} denotes the maximal element. For any
y,{displaystyle y,} and
holds.[3]
f-rings[edit]
An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring
(A,≤){displaystyle (A,leq )}x∧y=0{displaystyle xwedge y=0} in which
0≤z{displaystyle 0leq z} [4] and
zx∧y=xz∧y=0{displaystyle zxwedge y=xzwedge y=0} imply that
x,y,z∈A.{displaystyle x,y,zin A.} 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[edit]
Let
X{displaystyle X}C(X){displaystyle {mathcal {C}}(X)} be a Hausdorff space, and
X.{displaystyle X.} be the space of all continuous, real-valued functions on
C(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
C(X){displaystyle {mathcal {C}}(X)}
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[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]
|xy|=|x||y|{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
(A,≤){displaystyle (A,leq )}x,y,z∈A.{displaystyle x,y,zin A.} is a commutative ordered ring, and
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
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