OpenMath

OpenMath is an extensible standard for representing the semantics of mathematical objects.

https://openmath.org

OpenMath Content Dictionary: setname2

Canonical URL:
http://www.openmath.org/cd/setname2.ocd
CD Base:
http://www.openmath.org/cd
CD File:
setname2.ocd
CD as XML Encoded OpenMath:
setname2.omcd
Defines:
A, Boolean, GFp, GFpn, H, QuotientField, Zm
Date:
2004-03-30
Version:
3 (Revision 1)
Review Date:
2017-12-31
Status:
experimental

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

  Author: OpenMath Consortium
  SourceURL: https://github.com/OpenMath/CDs

This CD defines some common sets of mathematics. 

Written by J.H. Davenport on 1999-04-18.
Revised to add Zm, GFp, GFpn on 1999-11-09.
Revised to add QuotientField and A on 1999-11-19.

Boolean

Role:
constant
Description:

This symbol represents the set of Booleans. That is the truth values, true and false.

Commented Mathematical property (CMP):
for all b in the booleans | (there exists an nb in the booleans | nb not= b implies nb = not b)
Formal Mathematical property (FMP):
∀   b . b ∈ Boolean ⇒ ∃   nb . nb ∈ Boolean ∧ nb ≠ b ∧ nb = ¬ b
Signatures:
sts

[Next: A] [Last: H] [Top]

A

Role:
constant
Description:

This symbol represents the set of algebraic numbers.

Commented Mathematical property (CMP):
The algebraic numbers are a proper subset of the reals
Formal Mathematical property (FMP):
A ⊂ R
Commented Mathematical property (CMP):
The rationals are a proper subset of the algebraic numbers
Formal Mathematical property (FMP):
Q ⊂ A
Signatures:
sts

[Next: Zm] [Previous: Boolean] [Top]

Zm

Role:
application
Description:

This symbol represents the set of integers modulo m, where m is not necessarily a prime. It takes one argument, the integer m.

Commented Mathematical property (CMP):
for all x in the integers modulo m | there exists an n such that n is an integer and n <= m and x^n = x
Formal Mathematical property (FMP):
∀   x . x ∈ Z m ⇒ ∃   n . n ∈ Z ∧ n ≤ m ∧ x n = x
Example:
The integers mod 12:
Z 12
Example:
The integers mod m:
Z m
Example:
4*5=8 in Z mod 12
4 ⁢ 5 = 8
Signatures:
sts

[Next: GFp] [Previous: A] [Top]

GFp

Role:
application
Description:

This symbol represents the finite field of integers modulo p, where p is a prime.

Commented Mathematical property (CMP):
x^p = x mod p
Formal Mathematical property (FMP):
x p = x
Signatures:
sts

[Next: GFpn] [Previous: Zm] [Top]

GFpn

Role:
application
Description:

This symbol represents the finite field with p^n elements, where p is a prime.

Example:
GF p = GF p 1
Signatures:
sts

[Next: QuotientField] [Previous: GFp] [Top]

QuotientField

Role:
application
Description:

This symbol represents the quotient field of any integral domain.

Example:
The rationals equals QuotientField(Integers)
Q = QuotientField ⁡ ( Z )
Commented Mathematical property (CMP):
R is a field iff QuotientField(R)=R
Formal Mathematical property (FMP):
R ∈ structure ⁡ ( Field ) ≡ ( QuotientField ⁡ ( R ) = R )
Signatures:
sts

[Next: H] [Previous: GFpn] [Top]

H

Role:
constant
Description:

This symbol represents the set of quaternions.

Commented Mathematical property (CMP):
1 is a quaternion and there exists i,j,k s.t. i,j,k are quaternions and i^2 = j^2 = k^2 = ijk = -1 with abs(i) not = abs(j) not = abs(k) not = 1 implies for all q, q a quaternion implies there exists r_0, r_1, r_2, r_3 reals s.t. q = r_0 + r_1*i + r_2*j + r_3*k
Formal Mathematical property (FMP):
1 ∈ H ∧ ∃   i , j , k . i ∈ H ∧ j ∈ H ∧ k ∈ H ∧ i 2 = - 1 ∧ j 2 = - 1 ∧ k 2 = - 1 ∧ i ⁢ j ⁢ k = - 1 ∧ | i | ≠ 1 ∧ | j | ≠ 1 ∧ | k | ≠ 1 ∧ ∀   q . q ∈ H ⇒ ∃   r 0 , r 1 , r 2 , r 3 . r 0 ∈ R ∧ r 1 ∈ R ∧ r 2 ∈ R ∧ r 3 ∈ R ∧ q = r 0 + r 1 ⁢ i + r 2 ⁢ j + r 3 ⁢ k
Example:
There exists a,b in the quaternions s.t. a*b neq b*a
∃   a , b . a ⁢ b ≠ b ⁢ a
Signatures:
sts

[First: Boolean] [Previous: QuotientField] [Top]