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<h1>MatricesForHomalg</h1>


<h2>Matrices for the homalg project</h2>

<p>
    2025.09-01</p>

<p>
    7 September 2025
  </p>

</div>
<p><b>
    Mohamed Barakat




  </b>
<br />Email: <span class="URL"><a href="mailto:mohamed.barakat@uni-siegen.de">mohamed.barakat@uni-siegen.de</a></span>
<br />Homepage: <span class="URL"><a href="https://mohamed-barakat.github.io">https://mohamed-barakat.github.io</a></span>
<br />Address: <br />Walter-Flex-Str. 3<br /> 57072 Siegen<br /> Germany<br />
</p><p><b>
    Markus Lange-Hegermann




  </b>
<br />Email: <span class="URL"><a href="mailto:markus.lange-hegermann@hs-owl.de">markus.lange-hegermann@hs-owl.de</a></span>
<br />Homepage: <span class="URL"><a href="https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/">https://www.th-owl.de/eecs/fachbereich/team/markus-lange-hegermann/</a></span>
<br />Address: <br />Markus Lange-Hegermann<br /> Hochschule Ostwestfalen-Lippe<br /> Liebigstraße 87<br /> 32657 Lemgo<br /> Germany<br />
</p><p><b>
    Martin Leuner




  </b>
<br />Email: <span class="URL"><a href="mailto:leuner@momo.math.rwth-aachen.de">leuner@momo.math.rwth-aachen.de</a></span>
<br />Homepage: <span class="URL"><a href="http://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php">http://wwwb.math.rwth-aachen.de/Mitarbeiter/leuner.php</a></span>
<br />Address: <br />Martin Leuner<br /> Lehrstuhl B fuer Mathematik, RWTH Aachen<br /> Templergraben 64<br /> 52062 Aachen<br /> Germany<br />
</p><p><b>
    Vinay Wagh




  </b>
<br />Email: <span class="URL"><a href="mailto:waghoba@gmail.com">waghoba@gmail.com</a></span>
<br />Homepage: <span class="URL"><a href="http://www.iitg.ernet.in/vinay.wagh/">http://www.iitg.ernet.in/vinay.wagh/</a></span>
<br />Address: <br />E-102, Department of Mathematics,<br /> Indian Institute of Technology Guwahati,<br /> Guwahati, Assam, India.<br /> PIN: 781 039.<br /> India<br />
</p>

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<div class="contents">
<h3>Contents<a id="contents" name="contents"></a></h3>

<div class="ContChap"><a href="chap1_mj.html#X7DFB63A97E67C0A1">1 <span class="Heading">Introduction</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X878AE2517B963434">1.1 <span class="Heading">What is the role of the <strong class="pkg">MatricesForHomalg</strong> package in the <strong class="pkg">homalg</strong> project?</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X81B1923E82145E72">1.1-1 <span class="Heading"><strong class="pkg">MatricesForHomalg</strong> provides ...</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X856DCAA4846FBB00">1.1-2 <span class="Heading"><strong class="pkg">homalg</strong> delegates ...</span></a>
</span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap1_mj.html#X808E7BA97C5EA311">1.1-3 <span class="Heading">The black box concept</span></a>
</span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap1_mj.html#X78DD800B83ABC621">1.2 <span class="Heading">This manual</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap2_mj.html#X8609DF5282514B96">2 <span class="Heading">Installation of the <strong class="pkg">MatricesForHomalg</strong> Package</span></a>
</div>
<div class="ContChap"><a href="chap3_mj.html#X81897F6082CACB59">3 <span class="Heading">Rings</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8252B2F483D80E41">3.1 <span class="Heading">Rings: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85E217C67DD633AB">3.1-1 IsHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81DC249883163C01">3.1-2 IsPreHomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80A410ED8500DA7E">3.1-3 IsHomalgRingElement</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8097E89E7B6EF731">3.1-4 IsHomalgInternalRingRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7C7962B97E6CDFE2">3.2 <span class="Heading">Rings: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78AC74CB802A8A49">3.2-1 HomalgRingOfIntegers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85D9DDE384304BAB"><code>3.2-2 \/</code></a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7D171A1C797E27C9">3.3 <span class="Heading">Rings: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C48437187E668F3">3.3-1 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F80A53387A0C23D">3.3-2 IsNonZeroRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84F3040687E68338">3.3-3 ContainsAField</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C337D0F8413FE38">3.3-4 IsRationalsForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86221E0E8416F1CF">3.3-5 IsFieldForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X805112347CF99F02">3.3-6 IsDivisionRingForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799A9A9F7A26C6B2">3.3-7 IsIntegersForHomalg</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8548FE4E8283ACC6">3.3-8 IsResidueClassRingOfTheIntegers</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F9F59B5857F19A3">3.3-9 IsBezoutRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79D8752F78215FC1">3.3-10 IsIntegrallyClosedDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X864BF29E7B5D3305">3.3-11 IsUniqueFactorizationDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86EF914787EB5572">3.3-12 IsKaplanskyHermite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86C625EF7E417AA6">3.3-13 IsDedekindDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X855E560A7F40B2BF">3.3-14 IsDiscreteValuationRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80E0C8B28039B8F0">3.3-15 IsFreePolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X850A0EAB7E017D5E">3.3-16 IsWeylRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EFB456286B4F9DB">3.3-17 IsLocalizedWeylRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86558C9F8474DA39">3.3-18 IsGlobalDimensionFinite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AE1C7297A66F116">3.3-19 IsLeftGlobalDimensionFinite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799A94467B8EC416">3.3-20 IsRightGlobalDimensionFinite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X81269E1881D45163">3.3-21 HasInvariantBasisProperty</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8758DFD57E83925D">3.3-22 IsLocal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AAF0A3178E23B09">3.3-23 IsSemiLocalRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EE2F1C187131E19">3.3-24 IsIntegralDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FEB8A337CC92955">3.3-25 IsHereditary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D4AC0177C6D85A8">3.3-26 IsLeftHereditary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DE025D781FEBD04">3.3-27 IsRightHereditary</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783ACC147A7F82AA">3.3-28 IsHermite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A33BCFE7B6C6817">3.3-29 IsLeftHermite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X830989817DC97403">3.3-30 IsRightHermite</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AA2911E802BE73D">3.3-31 IsNoetherian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7803DB3A7E6689B6">3.3-32 IsLeftNoetherian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78A93EFA7B677CED">3.3-33 IsRightNoetherian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8373421F7E085763">3.3-34 IsCohenMacaulay</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83CBA38E81DC4A72">3.3-35 IsGorenstein</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E7AEFBE7801F196">3.3-36 IsKoszul</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AF81F6383F5CFCA">3.3-37 IsArtinian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E000F5780A17602">3.3-38 IsLeftArtinian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C34A319827FFDDB">3.3-39 IsRightArtinian</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8290570679F86CE8">3.3-40 IsOreDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8528CA397BC76826">3.3-41 IsLeftOreDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FC7E8317BF9B9CE">3.3-42 IsRightOreDomain</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85F1485F840E2354">3.3-43 IsPrincipalIdealRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BF4EFB67DCEBF6D">3.3-44 IsLeftPrincipalIdealRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X83858198873F7760">3.3-45 IsRightPrincipalIdealRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CF02C4785F0EAB5">3.3-46 IsRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7FB92D467B9B6707">3.3-47 IsFiniteFreePresentationRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B0EE3BF8402793B">3.3-48 IsLeftFiniteFreePresentationRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X839A82AC7D0D7BA1">3.3-49 IsRightFiniteFreePresentationRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8491CBBE862D4FFB">3.3-50 IsSimpleRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X847DEBCF872F5175">3.3-51 IsSemiSimpleRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X842C9ABA807DB431">3.3-52 IsSuperCommutative</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X803259617B5F89AE">3.3-53 BasisAlgorithmRespectsPrincipalIdeals</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X781617F678CC0BA8">3.3-54 AreUnitsCentral</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X85B6710082984863">3.3-55 IsMinusOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A0A3A927BE3F352">3.3-56 IsMonic</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X785EF83B8054D2FF">3.3-57 IsMonicUptoUnit</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X811A01D5803ADCA3">3.3-58 IsLeftRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E99731F83A41777">3.3-59 IsRightRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80A3294C834D8F21">3.3-60 IsRegular</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X867290E7847A5101">3.4 <span class="Heading">Rings: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8066502785A109B8">3.4-1 Inverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7AFD26D480AA9323">3.4-2 homalgTable</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X816D807781E8F854">3.4-3 RingElementConstructor</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E5426C67AA9A6E5">3.4-4 TypeOfHomalgMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80504BE983BD1A70">3.4-5 ConstructorForHomalgMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799B5F797F809EE5">3.4-6 Zero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84701329860750C3">3.4-7 One</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X810D03AA827BD128">3.4-8 MinusOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7CC4312578DC42B6">3.4-9 ProductOfIndeterminates</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7DF4F71C86835DCF">3.4-10 RationalParameters</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80D585E1793D4552">3.4-11 IndeterminatesOfPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84CE78E379A34C56">3.4-12 RelativeIndeterminatesOfPolynomialRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F4A050A87C042E5">3.4-13 IndeterminateCoordinatesOfRingOfDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X821FCC287E4FB92F">3.4-14 RelativeIndeterminateCoordinatesOfRingOfDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78776EBA7DC179B4">3.4-15 IndeterminateDerivationsOfRingOfDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8522A7987C6483ED">3.4-16 RelativeIndeterminateDerivationsOfRingOfDerivations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C15E6647945C0E3">3.4-17 IndeterminateAntiCommutingVariablesOfExteriorRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C63673A80911044">3.4-18 RelativeIndeterminateAntiCommutingVariablesOfExteriorRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7BBEF7097B459D33">3.4-19 IndeterminatesOfExteriorRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8235D10781BE8003">3.4-20 CoefficientsRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X789CF8B778A0C58D">3.4-21 KrullDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8735C56B7BEBC86E">3.4-22 LeftGlobalDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E6C5B5781EF78C5">3.4-23 RightGlobalDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D511B3E7A50AB2A">3.4-24 GlobalDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X792D56C278E346B1">3.4-25 GeneralLinearRank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79BCB23D873268CB">3.4-26 ElementaryRank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X822907CB7919EEF2">3.4-27 StableRank</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X826BE1E87EE023B2">3.4-28 AssociatedGradedRing</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7DDAB86C7A7FEDA9">3.5 <span class="Heading">Rings: Operations and Functions</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap4_mj.html#X7B222197819984A6">4 <span class="Heading">Ring Maps</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7B99B8F5780E84C3">4.1 <span class="Heading">Ring Maps:  Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7E084D947E3AEFE6">4.1-1 IsHomalgRingMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87DB79AF83F17FB6">4.1-2 IsHomalgRingSelfMap</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7DFD1CBA83E63737">4.1-3 IsHomalgRingMapRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X8717AEFB7BAC63F7">4.2 <span class="Heading">Ring Maps: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7F21AB318507FF83">4.2-1 RingMap</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X85DA972D8701BC7C">4.3 <span class="Heading">Ring Maps: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X8555A4DF84C9165B">4.3-1 IsMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X832893897FD3744D">4.3-2 IsIdentityMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X87F79EA381E3E34F">4.3-3 IsMonomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X849F620C824F4078">4.3-4 IsEpimorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X82B9422D7B01BA4A">4.3-5 IsIsomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X790E34C5802D0F54">4.3-6 IsAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7EBF1DD67BD0758F">4.4 <span class="Heading">Ring Maps: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X83678DEC78394702">4.4-1 Source</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7EBE68567900396A">4.4-2 Range</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X7C4F3F0F82C6EB88">4.4-3 DegreeOfMorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap4_mj.html#X785155EE844A98BD">4.4-4 CoordinateRingOfGraph</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html#X7C7401BA7E2221CB">4.5 <span class="Heading">Ring Maps: Operations and Functions</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chap5_mj.html#X812CCAB278643A59">5 <span class="Heading">Matrices</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X78C552687FF14479">5.1 <span class="Heading">Matrices: Category and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B68E1057F5F011F">5.1-1 IsHomalgMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7FE94FC47F460E35">5.1-2 IsHomalgInternalMatrixRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7977387186436CDF">5.2 <span class="Heading">Matrices: Constructors</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86D290B084AC6638">5.2-1 HomalgInitialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7CB77009868D369A">5.2-2 HomalgInitialIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8309EB7B86953A23">5.2-3 HomalgZeroMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83266B9D7BE740D8">5.2-4 HomalgIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D2E3472879E28AB">5.2-5 HomalgVoidMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X864ACCB08094F0B7">5.2-6 HomalgMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8246E1D17F96DAE7">5.2-7 HomalgMatrixListList</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B127B5584CD012D">5.2-8 HomalgRowVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X871AF271843FF2B5">5.2-9 HomalgColumnVector</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X872D39C678D0C4AE">5.2-10 HomalgDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81225377833C4644"><code>5.2-11 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C4E49D287011DCD">5.2-12 CoercedMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7D92ECFC8030CF40">5.3 <span class="Heading">Matrices: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X858B5AF57D5BC90A">5.3-1 IsZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X814D78347858EC13">5.3-2 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7813653578F174AB">5.3-3 IsUnitFree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8612CB4A82D6D79E">5.3-4 IsPermutationMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EEE3E9780EBA607">5.3-5 IsSpecialSubidentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8672364D79EBCC5D">5.3-6 IsSubidentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EF95CAD78BDE12F">5.3-7 IsLeftRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87C369D27D6AAF68">5.3-8 IsRightRegular</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X856E1D217A47EE8C">5.3-9 IsInvertibleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A4FA27C80BC42D1">5.3-10 IsLeftInvertibleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E43FDE57E8449B6">5.3-11 IsRightInvertibleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7BFC9266823F2C15">5.3-12 IsEmptyMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EEC8E768178696E">5.3-13 IsDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7848E6A0783B7428">5.3-14 IsScalarMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8740E71C799C0BCC">5.3-15 IsUpperTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X853A5B988306DBFE">5.3-16 IsLowerTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7976C42B7FA905EC">5.3-17 IsStrictUpperTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B0C78AF8056D650">5.3-18 IsStrictLowerTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81A2C3F67C99A3C2">5.3-19 IsUpperStairCaseMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B3A5DE1860373F0">5.3-20 IsLowerStairCaseMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7BAAE75A8660D7A5">5.3-21 IsTriangularMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F520F89821A8602">5.3-22 IsBasisOfRowsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D46613983DC5302">5.3-23 IsBasisOfColumnsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86445AD281024339">5.3-24 IsReducedBasisOfRowsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E6BB540865C0344">5.3-25 IsReducedBasisOfColumnsMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E6E51517822CB3F">5.3-26 IsInitialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EE624707ACEC26E">5.3-27 IsInitialIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X802794217F56DE51">5.3-28 IsVoidMatrix</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X86F766077C89558F">5.4 <span class="Heading">Matrices: Attributes</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C72971F7D0CA3C8">5.4-1 NumberRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X847D45BF7F2BC67C">5.4-2 NumberColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83045F6F82C180E1">5.4-3 DeterminantMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X828225E0857B1FDA">5.4-4 ZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X870D761F7AB96D12">5.4-5 ZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7991ED337C73065A">5.4-6 NonZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F335DCB7B8781E4">5.4-7 NonZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B7A073D7E1FAEA4">5.4-8 PositionOfFirstNonZeroEntryPerRow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83B389A97A703E42">5.4-9 PositionOfFirstNonZeroEntryPerColumn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X862841E68674FA2A">5.4-10 RowRankOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C61862E81CABD51">5.4-11 ColumnRankOfMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EFCE38281AE60F9">5.4-12 LeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87614CA48493B63F">5.4-13 RightInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7809E0507E882674">5.4-14 CoefficientsOfUnreducedNumeratorOfHilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7938E13A7EF4ADB1">5.4-15 CoefficientsOfNumeratorOfHilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X781E2CDB8743B1C6">5.4-16 UnreducedNumeratorOfHilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C44039382DD5D91">5.4-17 NumeratorOfHilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7B93B7D082A50E61">5.4-18 HilbertPoincareSeries</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84299BAB807A1E13">5.4-19 HilbertPolynomial</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7BC36CC67CB09858">5.4-20 AffineDimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87C428A079000336">5.4-21 AffineDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X82A1B55879AB1742">5.4-22 ProjectiveDegree</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X791B772A7E368A88">5.4-23 ConstantTermOfHilbertPolynomialn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X835972A77F02C5BB">5.4-24 MatrixOfSymbols</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X80FA5AE87E8591BC">5.5 <span class="Heading">Matrices: Operations and Functions</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81BBF79C79C3B6DF">5.5-1 HomalgRing</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7FBAA11B8008D936">5.5-2 LeftInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7AAD17D47839BCAE">5.5-3 RightInverse</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A7E42C179142727">5.5-4 LeftInverseLazy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7FA3E7617EED7E1E">5.5-5 RightInverseLazy</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X800FA81F7C42BFEA">5.5-6 Involution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D0D35B582D9C0B0">5.5-7 TransposedMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7CF5CE79796001F6">5.5-8 CertainRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8256AF2A840B19C4">5.5-9 CertainColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D6D0BDF854C9EBC">5.5-10 UnionOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7FF9661D85EC46B1">5.5-11 UnionOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E61390E79B663E8">5.5-12 ConvertRowToMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X853EA6C87EDDF6EF">5.5-13 ConvertColumnToMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7ED5C86379A647F2">5.5-14 ConvertMatrixToRow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84C7C1D07DB6FBAA">5.5-15 ConvertMatrixToColumn</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7C7830BE847D84B4">5.5-16 DiagMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7CDA5D848468A0AA">5.5-17 KroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7ECF744B7DE82BED">5.5-18 DualKroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D1A074278B415BE"><code>5.5-19 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X87C773DA85B21ADF"><code>5.5-20 \+</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X784B57617B24208C"><code>5.5-21 \-</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F5961D78754157B"><code>5.5-22 \*</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E2074A77AFF518A"><code>5.5-23 \=</code></a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85887BBB86F0A08B">5.5-24 GetColumnIndependentUnitPositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X824AB44184DD63B0">5.5-25 GetRowIndependentUnitPositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A1969A17979FC49">5.5-26 GetUnitPosition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X781B1C0C80529B09">5.5-27 Eliminate</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X80ADBE0D82CC6E85">5.5-28 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X868CDA327D6C8DDC">5.5-29 BasisOfColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F851EC7861170D1">5.5-30 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86C97DBB787BAD6D">5.5-31 DecideZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86ECEA9B7A4AE578">5.5-32 SyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86504B757F6DC990">5.5-33 SyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84A93458804F16F6">5.5-34 SyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D3FC0CE7B63AAF1">5.5-35 SyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X82E0FF517DC38040">5.5-36 ReducedBasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84CED11F7A633BDA">5.5-37 ReducedBasisOfColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DE458D67B9B85BF">5.5-38 ReducedSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8699114D7A865C11">5.5-39 ReducedSyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D9DEC6081AF0003">5.5-40 BasisOfRowsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7BBC885F7C24DEC2">5.5-41 BasisOfColumnsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8513963C84A9F8CB">5.5-42 DecideZeroRowsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A06BF7779830815">5.5-43 DecideZeroColumnsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81ABDA3E7D94C661">5.5-44 BasisOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X83A5B51980FFDE53">5.5-45 BasisOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85C980288304B4AC">5.5-46 DecideZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86C93ABD857447F8">5.5-47 SyzygiesOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X80325CAD7CE56F4F">5.5-48 SyzygiesOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86A798D4850BF9E8">5.5-49 ReducedSyzygiesOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8766BBD578557D15">5.5-50 ReducedSyzygiesOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X850AEC9C7C00AFF5">5.5-51 RightDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D0EAF527F8514E0">5.5-52 LeftDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A8546EA87E3AE67">5.5-53 RightDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86CEB1FC7C358777">5.5-54 LeftDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X86DFDD25824E2F35">5.5-55 SafeRightDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85B64BD47F0379C5">5.5-56 SafeLeftDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7F6BAF8E7F2343EA">5.5-57 UniqueRightDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8788CB987A8A18A7">5.5-58 UniqueLeftDivide</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X82B2C4987D6D5BD3">5.5-59 GenerateSameRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X867A947682754A9A">5.5-60 GenerateSameColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X81CDBBBE878DC1E5">5.5-61 SimplifyHomalgMatrixByLeftAndRightMultiplicationWithInvertibleMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X85DA10C07E7E5A3D">5.5-62 SimplifyHomalgMatrixByLeftMultiplicationWithInvertibleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X79C124318588F37F">5.5-63 SimplifyHomalgMatrixByRightMultiplicationWithInvertibleMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7EEFD7887E96714F">5.5-64 CoefficientsWithGivenMonomials</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chap6_mj.html#X8163F0658017F220">6 <span class="Heading">Ring Relations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7EB7C20C78788C69">6.1 <span class="Heading">Ring Relations: Categories and Representations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7D50E3AD82087AE6">6.1-1 IsHomalgRingRelations</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7DECADD683403C65">6.1-2 IsHomalgRingRelationsAsGeneratorsOfLeftIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X78746A217FEEB058">6.1-3 IsHomalgRingRelationsAsGeneratorsOfRightIdeal</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X86CA83A081B8C8EA">6.1-4 IsRingRelationsRep</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X81D1405F81B86E4B">6.2 <span class="Heading">Ring Relations: Constructors</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7FFB5DE07BB77319">6.3 <span class="Heading">Ring Relations: Properties</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X835DF250790EF863">6.3-1 CanBeUsedToDecideZero</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap6_mj.html#X7B9398827AEEA2E6">6.3-2 IsInjectivePresentation</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X849ED71B8164D1C2">6.4 <span class="Heading">Ring Relations: Attributes</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7ABFB8F982EBD7F8">6.5 <span class="Heading">Ring Relations: Operations and Functions</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapA_mj.html#X7CB422647C7DD289">A <span class="Heading">The Basic Matrix Operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X810454AB85D336F5">A.1 <span class="Heading">Main</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X8435BB2E7A819478">A.2 <span class="Heading">Effective</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7B1023F47EAB7A97">A.3 <span class="Heading">Relative</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapA_mj.html#X7A0B865E7E70DB3D">A.4 <span class="Heading">Reduced</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapB_mj.html#X7B2993CB7B012115">B <span class="Heading">The Matrix Tool Operations</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapB_mj.html#X7988F0AF7D87FD23">B.1 <span class="Heading">The Tool Operations <em>without</em> a Fallback Method</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7DBA33F083A317B5">B.1-1 InitialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X84179BE87E7DCE76">B.1-2 InitialIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X785390E38396CAEB">B.1-3 ZeroMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X87BFF3567DEEBEF4">B.1-4 IdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X85884C3178473521">B.1-5 Involution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7AD2EEE680DF472B">B.1-6 TransposedMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7B6FC3267CD9EE9D">B.1-7 CertainRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X78EADFC67D17CF04">B.1-8 CertainColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7DEB535782A3323E">B.1-9 UnionOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X86C345CE82AAB220">B.1-10 UnionOfRowsPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7DF5DB55836D13A7">B.1-11 UnionOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X8092789C87E37020">B.1-12 UnionOfColumnsPair</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X86C5B86981FA1F9A">B.1-13 DiagMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X82202A6A7FAB7174">B.1-14 KroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X87E0747D7FEEAC76">B.1-15 DualKroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X828F8C7785EEC3D1">B.1-16 MulMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7B0B12F080A90039">B.1-17 AddMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7FE11AA27AE7D2D7">B.1-18 SubMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7D491D957E63C3A4">B.1-19 Compose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X849BB912798A01EB">B.1-20 IsZeroMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7F4D7FAF821DA1C2">B.1-21 NumberRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7DFA534B7AFA2E17">B.1-22 NumberColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X80A573257D7F2E1A">B.1-23 Determinant</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X8450E904787CBD35">B.1-24 CoefficientsWithGivenMonomials</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapB_mj.html#X7912E42C81296637">B.2 <span class="Heading">The Tool Operations with a Fallback Method</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7871FE5478BFC167">B.2-1 AreEqualMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X80C1856D82172268">B.2-2 IsIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7B6420E88418316B">B.2-3 IsDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X872B70367F412945">B.2-4 ZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7A469E6D7EA63BB6">B.2-5 ZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7BCBACDB79C96FBF">B.2-6 GetColumnIndependentUnitPositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X855C57B6822E7A98">B.2-7 GetRowIndependentUnitPositions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X876495AA79063CDE">B.2-8 GetUnitPosition</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X7F40B57079CF80ED">B.2-9 PositionOfFirstNonZeroEntryPerRow</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapB_mj.html#X833B384278492266">B.2-10 PositionOfFirstNonZeroEntryPerColumn</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapC_mj.html#X8222352C78A19214">C <span class="Heading">Logic Subpackages</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapC_mj.html#X86BB747287348853">C.1 <span class="Heading"><strong class="pkg">LIRNG</strong>: Logical Implications for Rings</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapC_mj.html#X7B2915F1867EE8D0">C.2 <span class="Heading"><strong class="pkg">LIMAP</strong>: Logical Implications for Ring Maps</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapC_mj.html#X799DA94B849ABF1E">C.3 <span class="Heading"><strong class="pkg">LIMAT</strong>: Logical Implications for Matrices</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapC_mj.html#X847B8AB5843231C2">C.4 <span class="Heading"><strong class="pkg">COLEM</strong>: Clever Operations for Lazy Evaluated Matrices</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7EEAADA6807A5A45">C.4-1 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7B619CA885024F0F">C.4-2 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7EADAA3180A84318">C.4-3 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X78CCA57B84E51834">C.4-4 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X8362669D87FD667B">C.4-5 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X84D72DF482F70AD5">C.4-6 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7928991E8768FA72">C.4-7 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X78D5359480EFC5AC">C.4-8 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X87BF7FD083D0EE88">C.4-9 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X852DCBD57A742FA5">C.4-10 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X835F6F2E7D590F3D">C.4-11 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7F35A61C8522A1B0">C.4-12 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7EDE6095820F8128">C.4-13 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7FD68F43831046B6">C.4-14 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X84F45FB4854A079C">C.4-15 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X78ADE5C879583E7B">C.4-16 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7B68797C7EA79B10">C.4-17 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X85971C16868BD83C">C.4-18 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X86F848318791595C">C.4-19 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X7F7682FC86F602C2">C.4-20 Eval</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapC_mj.html#X848FE4F07BAF89DB">C.4-21 Eval</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapD_mj.html#X7F3BA9AE7A0D245D">D <span class="Heading">The subpackage <strong class="pkg">ResidueClassRingForHomalg</strong> as a sample ring package</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapD_mj.html#X84978AF3878A8375">D.1 <span class="Heading">The Mandatory Basic Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7AB980C5791BA204">D.1-1 BasisOfRowModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7F2B3332793FACA3">D.1-2 BasisOfColumnModule</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X83E072E1790A7D38">D.1-3 DecideZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X841426A87A1A20E4">D.1-4 DecideZeroColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X80F4836F7F175B12">D.1-5 SyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7899768C8304A59E">D.1-6 SyzygiesGeneratorsOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X78BC2E8E7A78CC82">D.1-7 BasisOfRowsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7D2E9D797877FFBD">D.1-8 BasisOfColumnsCoeff</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7F10DC697D2B828D">D.1-9 DecideZeroRowsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7B68B9F27BC02520">D.1-10 DecideZeroColumnsEffectively</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7DF62C5D7D2E6A6E">D.1-11 RelativeSyzygiesGeneratorsOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X852F9FD8837D97A5">D.1-12 RelativeSyzygiesGeneratorsOfColumns</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapD_mj.html#X83E14F457ADC297D">D.2 <span class="Heading">The Mandatory Tool Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X81CD6BAB7CA73AFC">D.2-1 InitialMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7D0F99857E280142">D.2-2 InitialIdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X80393225841391E7">D.2-3 ZeroMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X811B306C81435D87">D.2-4 IdentityMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7B322C637FC26E2D">D.2-5 Involution</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X816CBEA6790E0C31">D.2-6 TransposedMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7C5B29A37B13A53D">D.2-7 CertainRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X84CBE51981BA2C77">D.2-8 CertainColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X8510678E8569799E">D.2-9 UnionOfRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X862F756A7FC0F0D4">D.2-10 UnionOfColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X802BEBF5790D4167">D.2-11 DiagMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X85F1DDDB864AF265">D.2-12 KroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7E996A50863FE76C">D.2-13 DualKroneckerMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7A5D229384E9D19C">D.2-14 MulMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X83E1AEC781AE1274">D.2-15 AddMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X79A1C9297BE0C09A">D.2-16 SubMat</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X875447A686949D59">D.2-17 Compose</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X78D7BABE806B82FA">D.2-18 IsZeroMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X85CE26418598FACE">D.2-19 NumberRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X79A76B1A7CB57518">D.2-20 NumberColumns</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X80831B287AB565BA">D.2-21 Determinant</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapD_mj.html#X7A537DB185A0F67C">D.3 <span class="Heading">Some of the Recommended Tool Operations</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X848EB509816E8A7D">D.3-1 AreEqualMatrices</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X80122FB3846A6BA5">D.3-2 IsOne</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X87B8E7137DC97A71">D.3-3 IsDiagonalMatrix</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7EFC928C7E59CEAE">D.3-4 ZeroRows</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapD_mj.html#X7E78F7D6796C7016">D.3-5 ZeroColumns</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapE_mj.html#X82D40F3183F4F259">E <span class="Heading">Debugging <strong class="pkg">MatricesForHomalg</strong></span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapE_mj.html#X8062637283DD739D">E.1 <span class="Heading">Increase the assertion level</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapE_mj.html#X81D8EB2A7CE587C6">E.2 <span class="Heading"><code class="code">Using homalgMode</code></span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chapE_mj.html#X7D07F29F7EB515EE">E.2-1 homalgMode</a></span>
</div></div>
</div>
<div class="ContChap"><a href="chapF_mj.html#X863882737DAD95A3">F <span class="Heading">Overview of the <strong class="pkg">MatricesForHomalg</strong> Package Source Code</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X87E0F36680867FA2">F.1 <span class="Heading">Rings, Ring Maps, Matrices, Ring Relations</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X7C4917CE80359953">F.2 <span class="Heading">The Low Level Algorithms</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X7B2BFFB8876C548C">F.3 <span class="Heading">Logical Implications for <strong class="pkg">MatricesForHomalg</strong> Objects</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X85A3964E7A98C065">F.4 <span class="Heading">The subpackage <strong class="pkg">ResidueClassRingForHomalg</strong></span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chapF_mj.html#X860DCFF383750919">F.5 <span class="Heading">The homalgTable for <strong class="pkg">GAP4</strong> built-in rings</span></a>
</span>
</div>
</div>
<div class="ContChap"><a href="chapBib_mj.html"><span class="Heading">References</span></a></div>
<div class="ContChap"><a href="chapInd_mj.html"><span class="Heading">Index</span></a></div>
<br />
</div>

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