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[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/partially-ordered-ring-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki12\/partially-ordered-ring-wikipedia\/","headline":"Partially ordered ring – Wikipedia","name":"Partially ordered ring – Wikipedia","description":"Ring with a compatible partial order In abstract algebra, a partially ordered ring is a ring (A, +, \u00b7), together","datePublished":"2018-04-06","dateModified":"2018-04-06","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki12\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/11\/book.png","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24112548985eab096493f73f838580442780b57f","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/24112548985eab096493f73f838580442780b57f","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/en\/wiki12\/partially-ordered-ring-wikipedia\/","wordCount":6707,"articleBody":"Ring with a compatible partial order In abstract algebra, a partially ordered ring is a ring (A, +, \u00b7), together with a compatible partial order, that is, a partial order \u2264{displaystyle ,leq ,} on the underlying set A that is compatible with the ring operations in the sense that it satisfies:x\u2264y\u00a0implies\u00a0x+z\u2264y+z{displaystyle xleq y{text{ implies }}x+zleq y+z}and 0\u2264x\u00a0and\u00a00\u2264y\u00a0imply that\u00a00\u2264x\u22c5y{displaystyle 0leq x{text{ and }}0leq y{text{ imply that }}0leq xcdot y}for all x,y,z\u2208A{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 (A,\u2264){displaystyle (A,leq )} where 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,\u2264){displaystyle (A,leq )} where \u2264{displaystyle ,leq ,} is additionally a total order.[1][2]An l-ring, or lattice-ordered ring, is a partially ordered ring (A,\u2264){displaystyle (A,leq )} where \u2264{displaystyle ,leq ,} is additionally a lattice order.Table of ContentsProperties[edit]f-rings[edit]Example[edit]Properties[edit]Formally verified results for commutative ordered rings[edit]See also[edit]References[edit]Further reading[edit]External links[edit]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} for which 0\u2264x,{displaystyle 0leq x,} also called the positive cone of the ring) is closed under addition and multiplication, that is, if P{displaystyle P} is the set of non-negative elements of a partially ordered ring, then P+P\u2286P{displaystyle P+Psubseteq P} and P\u22c5P\u2286P.{displaystyle Pcdot Psubseteq P.} Furthermore, P\u2229(\u2212P)={0}.{displaystyle Pcap (-P)={0}.}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\u2286A{displaystyle Ssubseteq A} is a subset of a ring A,{displaystyle A,} and:0\u2208S{displaystyle 0in S}S\u2229(\u2212S)={0}{displaystyle Scap (-S)={0}}S+S\u2286S{displaystyle S+Ssubseteq S}S\u22c5S\u2286S{displaystyle Scdot Ssubseteq S}then the relation \u2264{displaystyle ,leq ,} where x\u2264y{displaystyle xleq y} if and only if y\u2212x\u2208S{displaystyle y-xin S} defines a compatible partial order on A{displaystyle A} (that is, (A,\u2264){displaystyle (A,leq )} is a partially ordered ring).[2]In any l-ring, the absolute value |x|{displaystyle |x|} of an element x{displaystyle x} can be defined to be x\u2228(\u2212x),{displaystyle xvee (-x),} where x\u2228y{displaystyle xvee y} denotes the maximal element. For any x{displaystyle x} and y,{displaystyle y,}|x\u22c5y|\u2264|x|\u22c5|y|{displaystyle |xcdot y|leq |x|cdot |y|}holds.[3]f-rings[edit]An f-ring, or Pierce\u2013Birkhoff ring, is a lattice-ordered ring (A,\u2264){displaystyle (A,leq )} in which x\u2227y=0{displaystyle xwedge y=0}[4] and 0\u2264z{displaystyle 0leq z} imply that zx\u2227y=xz\u2227y=0{displaystyle zxwedge y=xzwedge y=0} for all x,y,z\u2208A.{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 X{displaystyle X} be a Hausdorff space, and C(X){displaystyle {mathcal {C}}(X)} be the space of all continuous, real-valued functions on X.{displaystyle X.} 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)}[fg](x)=f(x)\u22c5g(x){displaystyle [fg](x)=f(x)cdot g(x)}[f\u2227g](x)=f(x)\u2227g(x).{displaystyle [fwedge g](x)=f(x)wedge g(x).}[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 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[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,\u2264){displaystyle (A,leq )} is a commutative ordered ring, and x,y,z\u2208A.{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 Ci\u00eancias. 28: 41\u201369.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 ppExternal links[edit]"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki12\/partially-ordered-ring-wikipedia\/#breadcrumbitem","name":"Partially ordered ring – Wikipedia"}}]}]