binopproperties Class Reference

Investigate the binary operators mathematical properties. More...

#include <binopproperties.h>

Collaboration diagram for binopproperties:

List of all members.

Public Member Functions
	binopproperties (uintc *mat_, uintc N_)
	mat_ is a pointer to a N_ by N_ matrix representing the binary operator.
uintc	inverseright (bool &found, uintc i, uintc e) const
	i*k == e then determine k.
uintc	inverseleft (bool &found, uintc k, uintc e) const
	i*k == e then determine i.
uintc	identityfind (bool &found) const
	Find the identity element.
uintc	matik (uintc i, uintc k) const
	Access the binary operator i*k.
boolc	isClosed () const
	Is the operator closed : each element 0 <= i*j < N .
boolc	isCommutative () const
	Is a*b == b*a?
boolc	isAssociative () const
	Is a*(b*c) == (a*b)*c?
boolc	isGroup () const
	Operator * is closed, associative, has identity and inverse on each element?
boolc	isGroupAbelian () const
	A group plus a*b==b*a.
Public Attributes
uintc *	mat
	Matrix of binary operator N by N.
uintc	N
	N operators.

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Detailed Description

Investigate the binary operators mathematical properties.

For example is the binary operator a group?

The binary matrix is passed in as a square N by N matrix. The matrix is made up of zero and positive integers. Let * be the binary operator.

Example

  cout << "Test if rotating a triange and flipping it form a group." << endl;
  uint mat[] = 
  {
    0,1,2,3,4,5,
    1,2,0,5,3,4,
    2,0,1,4,5,3,
    3,4,5,0,1,2,
    4,5,3,2,0,1,
    5,3,4,1,2,0
  };

  binopproperties gt(mat,6);
  cout << "isGroup()=" << gt.isGroup() << endl;

Definition at line 36 of file binopproperties.h.

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Constructor & Destructor Documentation

binopproperties::binopproperties	(	uintc *	mat_,
		uintc	N_
	)			[inline]

mat_ is a pointer to a N_ by N_ matrix representing the binary operator.

Definition at line 42 of file binopproperties.h.

    : mat(mat_), N(N_) {}

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Member Function Documentation

uintc binopproperties::identityfind

(

bool &

found

)

const

Find the identity element.

Definition at line 5 of file binopproperties.cpp.

References matik(), and N.

Referenced by isGroup(), binoppropertiestest::test04(), and binoppropertiestest::test05().

{
  uint k;
  for (uint i=0; i<N; ++i)
  {
    found=true;
    for (k=0; k<N; ++k)
    {
      if ((matik(i,k)==k) && (matik(k,i)==k))
        continue;

      found=false;
      k=N;
    }

    if (found)
      return i;
  }

  return 0;
}

uintc binopproperties::inverseleft	(	bool &	found,
		uintc	k,
		uintc	e
	)			const

i*k == e then determine i.

Definition at line 148 of file binopproperties.cpp.

Referenced by isGroup().

{
  found=true;
  for (uint w=0; w<N; ++w)
  {
    if (matik(w,k)==e)
      return w;
  }

  found=false;
  return 0;
}

uintc binopproperties::inverseright	(	bool &	found,
		uintc	i,
		uintc	e
	)			const

i*k == e then determine k.

Definition at line 130 of file binopproperties.cpp.

Referenced by isGroup().

{
  found=true;
  for (uint w=0; w<N; ++w)
  {
    if (matik(i,w)==e)
      return w;
  }

  found=false;
  return 0;
}

boolc binopproperties::isAssociative

(

)

const

Is a*(b*c) == (a*b)*c?

Definition at line 98 of file binopproperties.cpp.

References matik(), and N.

Referenced by isGroup(), binoppropertiestest::test04(), and binoppropertiestest::test05().

{
  uint i;
  uint k;
  uint w;

  uint a1;
  uint a2;
  for ( i=0; i<N; ++i)
  {
    for (k=0; k<N; ++k)
    {
      for (w=0; w<N; ++w)
      {
        a2 = matik(k,w);
        a2 = matik(i,a2);

        a1 = matik(i,k);
        a1 = matik(a1,w);

        if (a2!=a1)
          return false;
      }
    }
  }

  return true;
}

bool const binopproperties::isClosed

(

)

const

Is the operator closed : each element 0 <= i*j < N .

Definition at line 27 of file binopproperties.cpp.

References matik(), and N.

Referenced by isGroup(), binoppropertiestest::test04(), and binoppropertiestest::test05().

{
  uint x;
  for (uint i=0; i<N; ++i)
  {
    for (uint k=0; k<N; ++k)
    {
      x = matik(i,k);
      if (x>=N)
        return false;
    }
  }

  return true;
}

boolc binopproperties::isCommutative

(

)

const

Is a*b == b*a?

Definition at line 43 of file binopproperties.cpp.

References matik(), and N.

Referenced by isGroupAbelian(), binoppropertiestest::test04(), and binoppropertiestest::test05().

{
  for (uint i=0; i<N; ++i)
  {
    for (uint k=0; k<N/2; ++k)
    {      
      if (matik(i,k)!=matik(k,i))
        return false;
    }
  }

  return true;
}

boolc binopproperties::isGroup

(

)

const

Operator * is closed, associative, has identity and inverse on each element?

Definition at line 57 of file binopproperties.cpp.

References identityfind(), inverseleft(), inverseright(), isAssociative(), isClosed(), and N.

Referenced by isGroupAbelian(), binoppropertiestest::test04(), and binoppropertiestest::test05().

{
  if (isClosed()==false)
    return false;

  if (isAssociative()==false)
    return false;

  bool found;
  uint id = identityfind(found);
  if (found==false)
    return false;

  for (uint i=0; i<N; ++i)
  {
    inverseright(found,i,id);
    if (found==false)
      return false;
  }

  for (uint i=0; i<N; ++i)
  {
    inverseleft(found,i,id);
    if (found==false)
      return false;
  }

  return true;
}

boolc binopproperties::isGroupAbelian

(

)

const

A group plus a*b==b*a.

Definition at line 87 of file binopproperties.cpp.

References isCommutative(), and isGroup().

{
  if (isGroup()==false)
    return false;

  if (isCommutative()==false)
    return false;

  return true;
}

uintc binopproperties::matik	(	uintc	i,
		uintc	k
	)			const [inline]

Access the binary operator i*k.

Definition at line 59 of file binopproperties.h.

References mat, and N.

Referenced by identityfind(), isAssociative(), isClosed(), and isCommutative().

    { assert(i<N); assert(k<N); return mat[i*N+k]; }

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Member Data Documentation

uintc* binopproperties::mat

Matrix of binary operator N by N.

Definition at line 46 of file binopproperties.h.

Referenced by matik().

uintc binopproperties::N

N operators.

Definition at line 48 of file binopproperties.h.

Referenced by identityfind(), isAssociative(), isClosed(), isCommutative(), isGroup(), and matik().

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The documentation for this class was generated from the following files:

• mathlib/binopproperties.h
• mathlib/binopproperties.cpp