OpenMath Content Dictionary: fieldname1
-
Canonical URL:
-
http://www.openmath.org/cd/fieldname1.ocd
-
CD Base:
-
http://www.openmath.org/cd
-
CD File:
-
fieldname1.ocd
-
CD as XML Encoded OpenMath:
-
fieldname1.omcd
-
Defines:
-
C, Q, R
-
Date:
- 2004-06-01
-
Version:
- 1
(Revision 1)
-
Review Date:
- 2006-06-01
-
Status:
- experimental
A CD of
functions for basic constructions in field theory.
Written by Arjeh M. Cohen 2004-02-25
-
Description:
-
This is a symbol representing the field of complex numbers.
-
Commented Mathematical property (CMP):
-
The field of complex numbers is (C, +,0,-,*,1,/), where +,-,*,/ are the standard
arithmetic operations.
-
Formal Mathematical property (FMP):
-
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS cd="relation1" name="eq"/>
<OMS cd="fieldname1" name="C"/>
<OMA><OMS cd="field1" name="field"/>
<OMS cd="setname1" name="C"/>
<OMS cd="arith1" name="plus"/>
<OMI>0</OMI>
<OMS cd="arith1" name="minus"/>
<OMS cd="arith1" name="times"/>
<OMI>1</OMI>
<OMS cd="arith1" name="divide"/>
</OMA>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<csymbol cd="fieldname1">C</csymbol>
<apply><csymbol cd="field1">field</csymbol>
<csymbol cd="setname1">C</csymbol>
<csymbol cd="arith1">plus</csymbol>
<cn type="integer">0</cn>
<csymbol cd="arith1">minus</csymbol>
<csymbol cd="arith1">times</csymbol>
<cn type="integer">1</cn>
<csymbol cd="arith1">divide</csymbol>
</apply>
</apply>
</math>
fieldname1.C = field1.field(setname1.C, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)
-
Signatures:
-
sts
[Next: R]
[Last: Q]
[Top]
|
-
Description:
-
This is a symbol representing the field of real numbers.
-
Commented Mathematical property (CMP):
-
The field of real numbers is (R, +,0,-,*,1,/), where +,-,*,/ are the standard
arithmetic operations.
-
Formal Mathematical property (FMP):
-
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS cd="relation1" name="eq"/>
<OMS cd="fieldname1" name="R"/>
<OMA><OMS cd="field1" name="field"/>
<OMS cd="setname1" name="R"/>
<OMS cd="arith1" name="plus"/>
<OMI>0</OMI>
<OMS cd="arith1" name="minus"/>
<OMS cd="arith1" name="times"/>
<OMI>1</OMI>
<OMS cd="arith1" name="divide"/>
</OMA>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<csymbol cd="fieldname1">R</csymbol>
<apply><csymbol cd="field1">field</csymbol>
<csymbol cd="setname1">R</csymbol>
<csymbol cd="arith1">plus</csymbol>
<cn type="integer">0</cn>
<csymbol cd="arith1">minus</csymbol>
<csymbol cd="arith1">times</csymbol>
<cn type="integer">1</cn>
<csymbol cd="arith1">divide</csymbol>
</apply>
</apply>
</math>
fieldname1.R = field1.field(setname1.R, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)
-
Signatures:
-
sts
[Next: Q]
[Previous: C]
[Top]
|
-
Description:
-
This is a symbol representing the field of rational numbers.
-
Commented Mathematical property (CMP):
-
The field of rational numbers is (Q, +,0,-,*,1,/), where +,-,*,/ are the standard
arithmetic operations.
-
Formal Mathematical property (FMP):
-
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS cd="relation1" name="eq"/>
<OMS cd="fieldname1" name="Q"/>
<OMA><OMS cd="field1" name="field"/>
<OMS cd="setname1" name="Q"/>
<OMS cd="arith1" name="plus"/>
<OMI>0</OMI>
<OMS cd="arith1" name="minus"/>
<OMS cd="arith1" name="times"/>
<OMI>1</OMI>
<OMS cd="arith1" name="divide"/>
</OMA>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<csymbol cd="fieldname1">Q</csymbol>
<apply><csymbol cd="field1">field</csymbol>
<csymbol cd="setname1">Q</csymbol>
<csymbol cd="arith1">plus</csymbol>
<cn type="integer">0</cn>
<csymbol cd="arith1">minus</csymbol>
<csymbol cd="arith1">times</csymbol>
<cn type="integer">1</cn>
<csymbol cd="arith1">divide</csymbol>
</apply>
</apply>
</math>
fieldname1.Q = field1.field(setname1.Q, arith1.plus, 0, arith1.minus, arith1.times, 1, arith1.divide)
-
Commented Mathematical property (CMP):
-
The carrier set of this field is the set of rational numbers.
-
Formal Mathematical property (FMP):
-
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS name="eq" cd="relation1"/>
<OMA><OMS name="carrier" cd="ring1"/>
<OMS name="Q" cd="fieldname1"/>
</OMA>
<OMS name="Q" cd="setname1"/>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<apply><csymbol cd="ring1">carrier</csymbol><csymbol cd="fieldname1">Q</csymbol></apply>
<csymbol cd="setname1">Q</csymbol>
</apply>
</math>
ring1.carrier(fieldname1.Q) = setname1.Q
-
Example:
-
<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0">
<OMA><OMS cd="relation1" name="eq"/>
<OMS cd="fieldname1" name="Q"/>
<OMA><OMS cd="field1" name="field"/>
<OMS cd="setname1" name="Q"/>
<OMS cd="arith1" name="plus"/>
<OMI>0</OMI>
<OMS cd="arith1" name="minus"/>
<OMS cd="arith1" name="times"/>
<OMI>1</OMI>
<OMBIND><OMS cd="fns1" name="lambda"/>
<OMBVAR> <OMV name="x"/> </OMBVAR>
<OMA><OMS cd="arith1" name="divide"/>
<OMI> 1 </OMI> <OMV name="x"/>
</OMA>
</OMBIND>
</OMA>
</OMA>
</OMOBJ>
<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply><csymbol cd="relation1">eq</csymbol>
<csymbol cd="fieldname1">Q</csymbol>
<apply><csymbol cd="field1">field</csymbol>
<csymbol cd="setname1">Q</csymbol>
<csymbol cd="arith1">plus</csymbol>
<cn type="integer">0</cn>
<csymbol cd="arith1">minus</csymbol>
<csymbol cd="arith1">times</csymbol>
<cn type="integer">1</cn>
<bind><csymbol cd="fns1">lambda</csymbol>
<bvar><ci>x</ci></bvar>
<apply><csymbol cd="arith1">divide</csymbol><cn type="integer">1</cn><ci>x</ci></apply>
</bind>
</apply>
</apply>
</math>
fieldname1.Q = field1.field(setname1.Q, arith1.plus, 0, arith1.minus, arith1.times, 1, fns1.lambda[$x -> 1 / $x])
-
Signatures:
-
sts
[First: C]
[Previous: R]
[Top]
|