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<p><a id="X7BE194BD79C972A3" name="X7BE194BD79C972A3"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X7BE194BD79C972A3">3 <span class="Heading">The Ring Table</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7B7C37BD80727239">3.1 <span class="Heading">An Example for a Ring Table - Singular</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CA9554E855D5032">3.1-1 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A0EDA3284F0832B">3.1-2 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A8574FE7B4DCE59">3.1-3 BasisOfColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X870A963687F2867F">3.1-4 BasisOfColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7834682C822FC188">3.1-5 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A7ADD857AAD8158">3.1-6 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8721416787B06D9B">3.1-7 DecideZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X781FC1367F5A2EB7">3.1-8 DecideZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85D2A3EA86796DD5">3.1-9 SyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78551C36859F7524">3.1-10 SyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X834F1AF179AE3F7F">3.1-11 SyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BFA32BA80762292">3.1-12 SyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F4661FD863E3EB4">3.1-13 BasisOfRowsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X874402C08793EDAD">3.1-14 BasisOfRowsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X796A404D7A059D47">3.1-15 BasisOfColumnsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A404EC486BAD561">3.1-16 BasisOfColumnsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8009817D78CF032B">3.1-17 DecideZeroRowsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80A26CC279614874">3.1-18 DecideZeroRowsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X833A271E83F7B7E6">3.1-19 DecideZeroColumnsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F239DE47B7EEA55">3.1-20 DecideZeroColumnsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87131F9182FD3B1C">3.1-21 RelativeSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BF27670874C5CE1">3.1-22 RelativeSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FEB3B37852B72D4">3.1-23 RelativeSyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78370E58780C91C2">3.1-24 RelativeSyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E533B0A8392EBC6">3.1-25 ReducedSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A6E97FA7F1FF73D">3.1-26 ReducedSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FF049FA85A2C82E">3.1-27 ReducedSyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AF938828299812C">3.1-28 ReducedSyzygiesGeneratorsOfColumns</a></span>
</div></div>
</div>

<h3>3 <span class="Heading">The Ring Table</span></h3>

<p><a id="X7B7C37BD80727239" name="X7B7C37BD80727239"></a></p>

<h4>3.1 <span class="Heading">An Example for a Ring Table - Singular</span></h4>

<p>todo: introductory text, mention: transposed matrices, the macros, refer to the philosophy</p>

<p><a id="X7CA9554E855D5032" name="X7CA9554E855D5032"></a></p>

<h5>3.1-1 BasisOfRowModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfRowModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">BasisOfRowModule</code> (<a href="chap3_mj.html#X7A0EDA3284F0832B"><span class="RefLink">3.1-2</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
BasisOfRowModule :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      "unknown_number_of_rows",
      NumberColumns( M ),
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = BasisOfRowModule(", M, ")" ],
      "need_command",
      "BasisOfModule"
    );

    return N;

  end,
</pre></div>

<p><a id="X7A0EDA3284F0832B" name="X7A0EDA3284F0832B"></a></p>

<h5>3.1-2 BasisOfRowModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfRowModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    BasisOfRowModule := "\n\
proc BasisOfRowModule (matrix M)\n\
{\n\
  return(std(M));\n\
}\n\n",
</pre></div>

<p><a id="X7A8574FE7B4DCE59" name="X7A8574FE7B4DCE59"></a></p>

<h5>3.1-3 BasisOfColumnModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfColumnModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">BasisOfColumnModule</code> (<a href="chap3_mj.html#X870A963687F2867F"><span class="RefLink">3.1-4</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
BasisOfColumnModule :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      NumberRows( M ),
      "unknown_number_of_columns",
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = BasisOfColumnModule(", M, ")" ],
      "need_command",
      "BasisOfModule"
    );

    return N;

  end,
</pre></div>

<p><a id="X870A963687F2867F" name="X870A963687F2867F"></a></p>

<h5>3.1-4 BasisOfColumnModule</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfColumnModule</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    BasisOfColumnModule := "\n\
proc BasisOfColumnModule (matrix M)\n\
{\n\
  return(Involution(BasisOfRowModule(Involution(M))));\n\
}\n\n",
</pre></div>

<p><a id="X7834682C822FC188" name="X7834682C822FC188"></a></p>

<h5>3.1-5 DecideZeroRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroRows</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">DecideZeroRows</code> (<a href="chap3_mj.html#X7A7ADD857AAD8158"><span class="RefLink">3.1-6</span></a>) inside the computer algebra system. The rows of <var class="Arg">B</var> must form a basis (see <code class="func">BasisOfRowModule</code> (<a href="chap3_mj.html#X7CA9554E855D5032"><span class="RefLink">3.1-1</span></a>)).</p>

<div class="example"><pre>
DecideZeroRows :=
  function( A, B )
    local N;

    N := HomalgVoidMatrix(
      NumberRows( A ),
      NumberColumns( A ),
      HomalgRing( A )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = DecideZeroRows(", A, B, ")" ],
      "need_command",
      "DecideZero"
    );

    return N;

  end,
</pre></div>

<p><a id="X7A7ADD857AAD8158" name="X7A7ADD857AAD8158"></a></p>

<h5>3.1-6 DecideZeroRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroRows</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    DecideZeroRows := "\n\
proc DecideZeroRows (matrix A, module B)\n\
{\n\
  attrib(B,\"isSB\",1);\n\
  return(reduce(A,B));\n\
}\n\n",
</pre></div>

<p><a id="X8721416787B06D9B" name="X8721416787B06D9B"></a></p>

<h5>3.1-7 DecideZeroColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroColumns</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">DecideZeroColumns</code> (<a href="chap3_mj.html#X781FC1367F5A2EB7"><span class="RefLink">3.1-8</span></a>) inside the computer algebra system. The columns of <var class="Arg">B</var> must form a basis (see <code class="func">BasisOfColumnModule</code> (<a href="chap3_mj.html#X7A8574FE7B4DCE59"><span class="RefLink">3.1-3</span></a>)).</p>

<div class="example"><pre>
DecideZeroColumns :=
  function( A, B )
    local N;

    N := HomalgVoidMatrix(
      NumberRows( A ),
      NumberColumns( A ),
      HomalgRing( A )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = DecideZeroColumns(", A, B, ")" ],
      "need_command",
      "DecideZero"
    );

    return N;

  end,
</pre></div>

<p><a id="X781FC1367F5A2EB7" name="X781FC1367F5A2EB7"></a></p>

<h5>3.1-8 DecideZeroColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroColumns</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    DecideZeroColumns := "\n\
proc DecideZeroColumns (matrix A, matrix B)\n\
{\n\
  return(Involution(DecideZeroRows(Involution(A),Involution(B))));\n\
}\n\n",
</pre></div>

<p><a id="X85D2A3EA86796DD5" name="X85D2A3EA86796DD5"></a></p>

<h5>3.1-9 SyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">SyzygiesGeneratorsOfRows</code> (<a href="chap3_mj.html#X78551C36859F7524"><span class="RefLink">3.1-10</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
SyzygiesGeneratorsOfRows :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      "unknown_number_of_rows",
      NumberRows( M ),
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = SyzygiesGeneratorsOfRows(", M, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X78551C36859F7524" name="X78551C36859F7524"></a></p>

<h5>3.1-10 SyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    SyzygiesGeneratorsOfRows := "\n\
proc SyzygiesGeneratorsOfRows (matrix M)\n\
{\n\
  return(SyzForHomalg(M));\n\
}\n\n",
</pre></div>

<p><a id="X834F1AF179AE3F7F" name="X834F1AF179AE3F7F"></a></p>

<h5>3.1-11 SyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">SyzygiesGeneratorsOfColumns</code> (<a href="chap3_mj.html#X7BFA32BA80762292"><span class="RefLink">3.1-12</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
SyzygiesGeneratorsOfColumns :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      NumberColumns( M ),
      "unknown_number_of_columns",
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = SyzygiesGeneratorsOfColumns(", M, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X7BFA32BA80762292" name="X7BFA32BA80762292"></a></p>

<h5>3.1-12 SyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    SyzygiesGeneratorsOfColumns := "\n\
proc SyzygiesGeneratorsOfColumns (matrix M)\n\
{\n\
  return(Involution(SyzForHomalg(Involution(M))));\n\
}\n\n",
</pre></div>

<p><a id="X7F4661FD863E3EB4" name="X7F4661FD863E3EB4"></a></p>

<h5>3.1-13 BasisOfRowsCoeff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfRowsCoeff</code>( <var class="Arg">M</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">BasisOfRowsCoeff</code> (<a href="chap3_mj.html#X874402C08793EDAD"><span class="RefLink">3.1-14</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
BasisOfRowsCoeff :=
  function( M, T )
    local v, N;

    v := homalgStream( HomalgRing( M ) )!.variable_name;

    N := HomalgVoidMatrix(
      "unknown_number_of_rows",
      NumberColumns( M ),
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, T, " = BasisOfRowsCoeff(", M, ")" ],
      "need_command",
      "BasisCoeff"
    );

    return N;

  end,
</pre></div>

<p><a id="X874402C08793EDAD" name="X874402C08793EDAD"></a></p>

<h5>3.1-14 BasisOfRowsCoeff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfRowsCoeff</code>( <var class="Arg">M</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    BasisOfRowsCoeff := """
proc BasisOfRowsCoeff (matrix M)
{
  matrix B = BasisOfRowModule(M);
  option(noredSB);
  matrix T = lift(M,B);
  option(redSB);
  return(B,T);
}

""",
</pre></div>

<p><a id="X796A404D7A059D47" name="X796A404D7A059D47"></a></p>

<h5>3.1-15 BasisOfColumnsCoeff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfColumnsCoeff</code>( <var class="Arg">M</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">BasisOfColumnsCoeff</code> (<a href="chap3_mj.html#X7A404EC486BAD561"><span class="RefLink">3.1-16</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
BasisOfColumnsCoeff :=
  function( M, T )
    local v, N;

    v := homalgStream( HomalgRing( M ) )!.variable_name;

    N := HomalgVoidMatrix(
      NumberRows( M ),
      "unknown_number_of_columns",
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, T, " = BasisOfColumnsCoeff(", M, ")" ],
      "need_command",
      "BasisCoeff"
    );

    return N;

  end,
</pre></div>

<p><a id="X7A404EC486BAD561" name="X7A404EC486BAD561"></a></p>

<h5>3.1-16 BasisOfColumnsCoeff</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ BasisOfColumnsCoeff</code>( <var class="Arg">M</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    BasisOfColumnsCoeff := """
proc BasisOfColumnsCoeff (matrix M)
{
  matrix B,T = BasisOfRowsCoeff(Involution(M));
  return(Involution(B),Involution(T));
}

""",
</pre></div>

<p><a id="X8009817D78CF032B" name="X8009817D78CF032B"></a></p>

<h5>3.1-17 DecideZeroRowsEffectively</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroRowsEffectively</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">DecideZeroRowsEffectively</code> (<a href="chap3_mj.html#X80A26CC279614874"><span class="RefLink">3.1-18</span></a>) inside the computer algebra system. The rows of <var class="Arg">B</var> must form a basis (see <code class="func">BasisOfRowModule</code> (<a href="chap3_mj.html#X7CA9554E855D5032"><span class="RefLink">3.1-1</span></a>)).</p>

<div class="example"><pre>
DecideZeroRowsEffectively :=
  function( A, B, T )
    local v, N;

    v := homalgStream( HomalgRing( A ) )!.variable_name;

    N := HomalgVoidMatrix(
      NumberRows( A ),
      NumberColumns( A ),
      HomalgRing( A )
    );

    homalgSendBlocking(
      [ "matrix ", N, T, " = DecideZeroRowsEffectively(", A, B, ")" ],
      "need_command",
      "DecideZeroEffectively"
    );

    return N;

  end,
</pre></div>

<p><a id="X80A26CC279614874" name="X80A26CC279614874"></a></p>

<h5>3.1-18 DecideZeroRowsEffectively</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroRowsEffectively</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    DecideZeroRowsEffectively := """
proc DecideZeroRowsEffectively (matrix A, module B)
{
  attrib(B,"isSB",1);
  matrix M = reduce(A,B);
  matrix T = lift(B,M-A);
  return(M,T);
}

""",
</pre></div>

<p><a id="X833A271E83F7B7E6" name="X833A271E83F7B7E6"></a></p>

<h5>3.1-19 DecideZeroColumnsEffectively</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroColumnsEffectively</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">DecideZeroColumnsEffectively</code> (<a href="chap3_mj.html#X7F239DE47B7EEA55"><span class="RefLink">3.1-20</span></a>) inside the computer algebra system. The columns of <var class="Arg">B</var> must form a basis (see <code class="func">BasisOfColumnModule</code> (<a href="chap3_mj.html#X7A8574FE7B4DCE59"><span class="RefLink">3.1-3</span></a>)).</p>

<div class="example"><pre>
DecideZeroColumnsEffectively :=
  function( A, B, T )
    local v, N;

    v := homalgStream( HomalgRing( A ) )!.variable_name;

    N := HomalgVoidMatrix(
      NumberRows( A ),
      NumberColumns( A ),
      HomalgRing( A )
    );

    homalgSendBlocking(
      [ "matrix ", N, T, " = DecideZeroColumnsEffectively(", A, B, ")" ],
      "need_command",
      "DecideZeroEffectively"
    );

    return N;

  end,
</pre></div>

<p><a id="X7F239DE47B7EEA55" name="X7F239DE47B7EEA55"></a></p>

<h5>3.1-20 DecideZeroColumnsEffectively</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DecideZeroColumnsEffectively</code>( <var class="Arg">A</var>, <var class="Arg">B</var>, <var class="Arg">T</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    DecideZeroColumnsEffectively := """
proc DecideZeroColumnsEffectively (matrix A, matrix B)
{
  matrix M,T = DecideZeroRowsEffectively(Involution(A),Involution(B));
  return(Involution(M),Involution(T));
}

""",
</pre></div>

<p><a id="X87131F9182FD3B1C" name="X87131F9182FD3B1C"></a></p>

<h5>3.1-21 RelativeSyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeSyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var>, <var class="Arg">M2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">RelativeSyzygiesGeneratorsOfRows</code> (<a href="chap3_mj.html#X7BF27670874C5CE1"><span class="RefLink">3.1-22</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
RelativeSyzygiesGeneratorsOfRows :=
  function( M, M2 )
    local N;

    N := HomalgVoidMatrix(
      "unknown_number_of_rows",
      NumberRows( M ),
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = RelativeSyzygiesGeneratorsOfRows(", M, M2, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X7BF27670874C5CE1" name="X7BF27670874C5CE1"></a></p>

<h5>3.1-22 RelativeSyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeSyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var>, <var class="Arg">M2</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    RelativeSyzygiesGeneratorsOfRows := "\n\
proc RelativeSyzygiesGeneratorsOfRows (matrix M1, matrix M2)\n\
{\n\
  return(modulo(M1, M2));\n\
}\n\n",
</pre></div>

<p><a id="X7FEB3B37852B72D4" name="X7FEB3B37852B72D4"></a></p>

<h5>3.1-23 RelativeSyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeSyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var>, <var class="Arg">M2</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">RelativeSyzygiesGeneratorsOfColumns</code> (<a href="chap3_mj.html#X78370E58780C91C2"><span class="RefLink">3.1-24</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
RelativeSyzygiesGeneratorsOfColumns :=
  function( M, M2 )
    local N;

    N := HomalgVoidMatrix(
      NumberColumns( M ),
      "unknown_number_of_columns",
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = RelativeSyzygiesGeneratorsOfColumns(", M, M2, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X78370E58780C91C2" name="X78370E58780C91C2"></a></p>

<h5>3.1-24 RelativeSyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RelativeSyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var>, <var class="Arg">M2</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    RelativeSyzygiesGeneratorsOfColumns := "\n\
proc RelativeSyzygiesGeneratorsOfColumns (matrix M1, matrix M2)\n\
{\n\
  return(Involution(RelativeSyzygiesGeneratorsOfRows(Involution(M1),Involution(M2))));\n\
}\n\n",
</pre></div>

<p><a id="X7E533B0A8392EBC6" name="X7E533B0A8392EBC6"></a></p>

<h5>3.1-25 ReducedSyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReducedSyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">ReducedSyzygiesGeneratorsOfRows</code> (<a href="chap3_mj.html#X7A6E97FA7F1FF73D"><span class="RefLink">3.1-26</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
ReducedSyzygiesGeneratorsOfRows :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      "unknown_number_of_rows",
      NumberRows( M ),
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = ReducedSyzygiesGeneratorsOfRows(", M, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X7A6E97FA7F1FF73D" name="X7A6E97FA7F1FF73D"></a></p>

<h5>3.1-26 ReducedSyzygiesGeneratorsOfRows</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReducedSyzygiesGeneratorsOfRows</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    ReducedSyzForHomalg := "\n\
proc ReducedSyzForHomalg (matrix M)\n\
{\n\
  return(matrix(nres(M,2)[2]));\n\
}\n\n",
    ReducedSyzygiesGeneratorsOfRows := "\n\
proc ReducedSyzygiesGeneratorsOfRows (matrix M)\n\
{\n\
  return(ReducedSyzForHomalg(M));\n\
}\n\n",
</pre></div>

<p><a id="X7FF049FA85A2C82E" name="X7FF049FA85A2C82E"></a></p>

<h5>3.1-27 ReducedSyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReducedSyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This is the entry of the <strong class="pkg">homalg</strong> table, which calls the corresponding macro <code class="func">ReducedSyzygiesGeneratorsOfColumns</code> (<a href="chap3_mj.html#X7AF938828299812C"><span class="RefLink">3.1-28</span></a>) inside the computer algebra system.</p>

<div class="example"><pre>
ReducedSyzygiesGeneratorsOfColumns :=
  function( M )
    local N;

    N := HomalgVoidMatrix(
      NumberColumns( M ),
      "unknown_number_of_columns",
      HomalgRing( M )
    );

    homalgSendBlocking(
      [ "matrix ", N, " = ReducedSyzygiesGeneratorsOfColumns(", M, ")" ],
      "need_command",
      "SyzygiesGenerators"
    );

    return N;

  end,
</pre></div>

<p><a id="X7AF938828299812C" name="X7AF938828299812C"></a></p>

<h5>3.1-28 ReducedSyzygiesGeneratorsOfColumns</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReducedSyzygiesGeneratorsOfColumns</code>( <var class="Arg">M</var> )</td><td class="tdright">( function )</td></tr></table></div>

<div class="example"><pre>
    ReducedSyzygiesGeneratorsOfColumns := "\n\
proc ReducedSyzygiesGeneratorsOfColumns (matrix M)\n\
{\n\
  return(Involution(ReducedSyzForHomalg(Involution(M))));\n\
}\n\n",
</pre></div>

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