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 ≤ {\displaystyle \,\leq \,} {\displaystyle \,\leq \,} on the underlying set A that is compatible with the ring operations in the sense that it satisfies: x ≤ y  implies  x + z ≤ y + z {\displaystyle x\leq y{\text{ implies }}x+z\leq y+z} {\displaystyle x\leq y{\text{ implies }}x+z\leq y+z} and 0 ≤ x  and  0 ≤ y  imply that  0 ≤ x ⋅ y {\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y} {\displaystyle 0\leq x{\text{ and }}0\leq y{\text{ imply that }}0\leq x\cdot y} for all x , y , z ∈ A {\displaystyle x,y,z\in A} {\displaystyle x,y,z\in 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 ( A , ≤ ) {\displaystyle (A,\leq )} {\displaystyle (A,\leq )} where A {\displaystyle A} {\displaystyle A}'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 (A,\leq )} where ≤ {\displaystyle \,\leq \,} {\displaystyle \,\leq \,} is additionally a total order.[1][2]

An l-ring, or lattice-ordered ring, is a partially ordered ring ( A , ≤ ) {\displaystyle (A,\leq )} {\displaystyle (A,\leq )} where ≤ {\displaystyle \,\leq \,} {\displaystyle \,\leq \,} 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 {\displaystyle x} {\displaystyle x} for which 0 ≤ x , {\displaystyle 0\leq x,} {\displaystyle 0\leq x,} also called the positive cone of the ring) is closed under addition and multiplication, that is, if P {\displaystyle P} {\displaystyle P} is the set of non-negative elements of a partially ordered ring, then P + P ⊆ P {\displaystyle P+P\subseteq P} {\displaystyle P+P\subseteq P} and P ⋅ P ⊆ P . {\displaystyle P\cdot P\subseteq P.} {\displaystyle P\cdot P\subseteq P.} Furthermore, P ∩ ( − P ) = { 0 } . {\displaystyle P\cap (-P)=\{0\}.} {\displaystyle P\cap (-P)=\{0\}.}

The mapping of the compatible partial order on a ring A {\displaystyle 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 S\subseteq A} {\displaystyle S\subseteq A} is a subset of a ring A , {\displaystyle A,} {\displaystyle A,} and:

  1. 0 ∈ S {\displaystyle 0\in S} {\displaystyle 0\in S}
  2. S ∩ ( − S ) = { 0 } {\displaystyle S\cap (-S)=\{0\}} {\displaystyle S\cap (-S)=\{0\}}
  3. S + S ⊆ S {\displaystyle S+S\subseteq S} {\displaystyle S+S\subseteq S}
  4. S ⋅ S ⊆ S {\displaystyle S\cdot S\subseteq S} {\displaystyle S\cdot S\subseteq S}

then the relation ≤ {\displaystyle \,\leq \,} {\displaystyle \,\leq \,} where x ≤ y {\displaystyle x\leq y} {\displaystyle x\leq y} if and only if y − x ∈ S {\displaystyle y-x\in S} {\displaystyle y-x\in S} defines a compatible partial order on A {\displaystyle A} {\displaystyle A} (that is, ( A , ≤ ) {\displaystyle (A,\leq )} {\displaystyle (A,\leq )} is a partially ordered ring).[2]

In any l-ring, the absolute value | x | {\displaystyle |x|} {\displaystyle |x|} of an element x {\displaystyle x} {\displaystyle x} can be defined to be x ∨ ( − x ) , {\displaystyle x\vee (-x),} {\displaystyle x\vee (-x),} where x ∨ y {\displaystyle x\vee y} {\displaystyle x\vee y} denotes the maximal element. For any x {\displaystyle x} {\displaystyle x} and y , {\displaystyle y,} {\displaystyle y,} | x ⋅ y | ≤ | x | ⋅ | y | {\displaystyle |x\cdot y|\leq |x|\cdot |y|} {\displaystyle |x\cdot y|\leq |x|\cdot |y|} holds.[3]

f-rings

An f-ring, or Pierce–Birkhoff ring, is a lattice-ordered ring ( A , ≤ ) {\displaystyle (A,\leq )} {\displaystyle (A,\leq )} in which x ∧ y = 0 {\displaystyle x\wedge y=0} {\displaystyle x\wedge y=0}[4] and 0 ≤ z {\displaystyle 0\leq z} {\displaystyle 0\leq z} imply that z x ∧ y = x z ∧ y = 0 {\displaystyle zx\wedge y=xz\wedge y=0} {\displaystyle zx\wedge y=xz\wedge y=0} for all x , y , z ∈ A . {\displaystyle x,y,z\in A.} {\displaystyle x,y,z\in A.} They were first introduced by Garrett Birkhoff and Richard S. Pierce in 1956, in a paper titled "Lattice-ordered rings",[5] 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

Let X {\displaystyle X} {\displaystyle X} be a Hausdorff space, and C ( X ) {\displaystyle {\mathcal {C}}(X)} {\displaystyle {\mathcal {C}}(X)} be the space of all continuous, real-valued functions on X . {\displaystyle X.} {\displaystyle X.} C ( X ) {\displaystyle {\mathcal {C}}(X)} {\displaystyle {\mathcal {C}}(X)} is an Archimedean f-ring with 1 under the following pointwise operations: [ f + g ] ( x ) = f ( x ) + g ( x ) {\displaystyle [f+g](x)=f(x)+g(x)} {\displaystyle [f+g](x)=f(x)+g(x)} [ f g ] ( x ) = f ( x ) ⋅ g ( x ) {\displaystyle [fg](x)=f(x)\cdot g(x)} {\displaystyle [fg](x)=f(x)\cdot g(x)} [ f ∧ g ] ( x ) = f ( x ) ∧ g ( x ) . {\displaystyle [f\wedge g](x)=f(x)\wedge g(x).} {\displaystyle [f\wedge g](x)=f(x)\wedge g(x).}[2]

From an algebraic point of view the rings C ( X ) {\displaystyle {\mathcal {C}}(X)} {\displaystyle {\mathcal {C}}(X)} are fairly rigid. For example, localisations, residue rings or limits of rings of the form C ( X ) {\displaystyle {\mathcal {C}}(X)} {\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

  • 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]
  • | x y | = | x | | y | {\displaystyle |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.[6]
  • 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]

See also

References

  1. Anderson, F. W. "Lattice-ordered rings of quotients". Canadian Journal of Mathematics. 17: 434–448. doi:10.4153/cjm-1965-044-7.
  2. 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. 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. ∧ {\displaystyle \wedge } {\displaystyle \wedge } denotes infimum.
  5. Birkhoff, G.; R. Pierce (1956). "Lattice-ordered rings". Anais da Academia Brasileira de Ciências. 28: 41–69.
  6. 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.

Further reading