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<span class="Heading" >Morphisms</span ></a>div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7DE206257C909BDE" >4 .1 <span class="Heading" >Morphisms: Categories and Representations</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7D0F89828196DFF0" >4 .1 -1 IsHomalgMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X81458CA5836D582F" >4 .1 -2 IsHomalgStaticMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7933C51A842ABA32" >4 .1 -3 IsHomalgEndomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X823580787F23EB10" >4 .1 -4 IsMorphismOfFinitelyGeneratedObjectsRep</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X84A97E897C74B492" >4 .1 -5 IsStaticMorphismOfFinitelyGeneratedObjectsRep</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X86A95A9B85D8B58B" >4 .2 <span class="Heading" >Morphisms: Constructors</span ></a>span >div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X7B0B60BD79756A00" >4 .3 <span class="Heading" >Morphisms: Properties</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F66120A814DC16B" >4 .3 -1 IsMorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B7206EC7F584F25" >4 .3 -2 IsGeneralizedMorphismWithFullDomain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7AD32A427B247366" >4 .3 -3 IsGeneralizedEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X83C68AEA7FE4AA29" >4 .3 -4 IsGeneralizedMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X83F05F467DA5EA4D" >4 .3 -5 IsGeneralizedIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X814D78347858EC13" >4 .3 -6 IsOne</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7CB5896082D29173" >4 .3 -7 IsIdempotent</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X78AD1FDD7F53932C" >4 .3 -8 IsMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8724CEF182DC4064" >4 .3 -9 IsEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7DFACF1F7D7F7EE9" >4 .3 -10 IsSplitMonomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X80A66EFA862E56BC" >4 .3 -11 IsSplitEpimorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7E07BBF57B92BA56" >4 .3 -12 IsIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F30E3D37E9D7F37" >4 .3 -13 IsAutomorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X806EEA4685A4A3F3" >4 .4 <span class="Heading" >Morphisms: Attributes</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7DE8173F80E07AB1" >4 .4 -1 Source </a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X829F76BB80BD55DB" >4 .4 -2 Range</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7F3927E287087B64" >4 .4 -3 CokernelEpi</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7D71AE8E838712D7" >4 .4 -4 CokernelNaturalGeneralizedIsomorphism</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X87C00FFB79FA93A8" >4 .4 -5 KernelSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X82672DB279FAEFCC" >4 .4 -6 KernelEmb</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X82FB6A4687E778D5" >4 .4 -7 ImageSubobject</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X85FA7C19800F72B2" >4 .4 -8 ImageObjectEmb</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X86E3E1BA7BCE4D66" >4 .4 -9 ImageObjectEpi</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X823682157C6B4D63" >4 .4 -10 MorphismAid</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X85F22F4084EA7D31" >4 .4 -11 InverseOfGeneralizedMorphismWithFullDomain</a></span >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X8500C49A784C8EDC" >4 .4 -12 DegreeOfMorphism</a></span >div ></div >div class="ContSect" ><span class="tocline" ><span class="nocss" > </span ><a href="chap4.html#X789623548056F7B7" >4 .5 <span class="Heading" >Morphisms: Operations and Functions</span ></a>span >div class="ContSSBlock" >span class="ContSS" ><br /><span class="nocss" > </span ><a href="chap4.html#X7B4F9EF27A241520" >4 .5 -1 ByASmallerPresentation</a></span >div ></div >div >4 <span class="Heading" >Morphisms</span ></h3>"X7DE206257C909BDE" name="X7DE206257C909BDE" ></a></p>4 .1 <span class="Heading" >Morphisms: Categories and Representations</span ></h4>"X7D0F89828196DFF0" name="X7D0F89828196DFF0" ></a></p>4 .1 -1 IsHomalgMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsHomalgMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong >-category which will include the <strong class="pkg" >GAP</strong >-categories <code class="func" >IsHomalgStaticMorphism</code > (<a href="chap4.html#X81458CA5836D582F" ><span class="RefLink" >4 .1 -2 </span ></a>) and <code class="func" >IsHomalgChainMorphism</code > (<a href="chap7.html#X7CB62E188027B7C5" ><span class="RefLink" >7 .1 1 </span ></a>). We need this <strong class="pkg" >GAP</strong >-category to be able to build complexes with *objects* being objects of <strong class="pkg" >homalg</strong > categories or again complexes. We need this GAP-category to be able to build chain morphisms with *morphisms* being morphisms of <strong class="pkg" >homalg</strong > categories or again chain morphisms. <br /> CAUTION: Never let <strong class="pkg" >homalg</strong > morphisms (which are not endomorphisms) be multiplicative elements!!</p>div class="example" ><pre >"IsHomalgMorphism" ,pre ></div >"X81458CA5836D582F" name="X81458CA5836D582F" ></a></p>4 .1 -2 IsHomalgStaticMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsHomalgStaticMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong >-category which will include the <strong class="pkg" >GAP</strong >-categories <code class="code" >IsHomalgMap</code >, etc. <br /> CAUTION: Never let homalg morphisms (which are not endomorphisms) be multiplicative elements!!</p>div class="example" ><pre >"IsHomalgStaticMorphism" ,pre ></div >"X7933C51A842ABA32" name="X7933C51A842ABA32" ></a></p>4 .1 -3 IsHomalgEndomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsHomalgEndomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( category )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong >-category which will include the <strong class="pkg" >GAP</strong >-categories <code class="code" >IsHomalgSelfMap</code >, <code class="func" >IsHomalgChainEndomorphism</code > (<a href="chap7.html#X853BD37084BFC602" ><span class="RefLink" >7 .1 -2 </span ></a>), etc. be multiplicative elements!!</p>div class="example" ><pre >"IsHomalgEndomorphism" ,pre ></div >"X823580787F23EB10" name="X823580787F23EB10" ></a></p>4 .1 -4 IsMorphismOfFinitelyGeneratedObjectsRep</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsMorphismOfFinitelyGeneratedObjectsRep</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( representation )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > representation of morphisms of finitley generated <strong class="pkg" >homalg</strong > objects.</p>strong class="pkg" >GAP</strong > category <code class="func" >IsHomalgMorphism</code > (<a href="chap4.html#X7D0F89828196DFF0" ><span class="RefLink" >4 .1 -1 </span ></a>).)</p>div class="example" ><pre >"IsMorphismOfFinitelyGeneratedObjectsRep" ,pre ></div >"X84A97E897C74B492" name="X84A97E897C74B492" ></a></p>4 .1 -5 IsStaticMorphismOfFinitelyGeneratedObjectsRep</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsStaticMorphismOfFinitelyGeneratedObjectsRep</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( representation )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >GAP</strong > representation of static morphisms of finitley generated <strong class="pkg" >homalg</strong > static objects.</p>strong class="pkg" >GAP</strong > category <code class="func" >IsHomalgStaticMorphism</code > (<a href="chap4.html#X81458CA5836D582F" ><span class="RefLink" >4 .1 -2 </span ></a>), which is a subrepresentation of the <strong class="pkg" >GAP</strong > representation <code class="func" >IsMorphismOfFinitelyGeneratedObjectsRep</code > (<a href="chap4.html#X823580787F23EB10" ><span class="RefLink" >4 .1 -4 </span ></a>).)</p>div class="example" ><pre >"IsStaticMorphismOfFinitelyGeneratedObjectsRep" ,pre ></div >"X86A95A9B85D8B58B" name="X86A95A9B85D8B58B" ></a></p>4 .2 <span class="Heading" >Morphisms: Constructors</span ></h4>"X7B0B60BD79756A00" name="X7B0B60BD79756A00" ></a></p>4 .3 <span class="Heading" >Morphisms: Properties</span ></h4>"X7F66120A814DC16B" name="X7F66120A814DC16B" ></a></p>4 .3 -1 IsMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>code class="code" >IsMorphism</code >=<code class="code" >true</code > means one of the following:</p>ul >li ><p>The property method <code class="code" >IsMorphism</code >(<var class="Arg" >phi</var >) was explicitly invoked by the user and it returned <code class="code" >true</code >, where prior to the invocation <code class="code" >HasIsMorphism</code >(<var class="Arg" >phi</var >) was <code class="code" >false</code >. The method is meant to check the integrity of the data structure at the time of it invocation. What this precisely means depends on the specific <strong class="pkg" >homalg</strong >-based package.</p>li >li ><p>The user has explicitly <code class="code" >SetIsMorphism</code >(<var class="Arg" >phi</var >, <code "code" >true</code >).</p>li >li ><p>The morphism <var class="Arg" >phi</var > is output of a categorical procedure where <code class="code" >IsMorphism</code > has become <code class="code" >true</code > for all morphisms in the input .</p>li >li ><p>The morphism <var class="Arg" >phi</var > is output of a categorical procedure which gurantees the integrity of the data structure of its output independent of its input .</p>li >ul >"X7B7206EC7F584F25" name="X7B7206EC7F584F25" ></a></p>4 .3 -2 IsGeneralizedMorphismWithFullDomain</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMorphismWithFullDomain</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >phi</var > is a generalized morphism.</p>"X7AD32A427B247366" name="X7AD32A427B247366" ></a></p>4 .3 -3 IsGeneralizedEpimorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedEpimorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >phi</var > is a generalized epimorphism.</p>"X83C68AEA7FE4AA29" name="X83C68AEA7FE4AA29" ></a></p>4 .3 -4 IsGeneralizedMonomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedMonomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >phi</var > is a generalized monomorphism.</p>"X83F05F467DA5EA4D" name="X83F05F467DA5EA4D" ></a></p>4 .3 -5 IsGeneralizedIsomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsGeneralizedIsomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>var class="Arg" >phi</var > is a generalized isomorphism.</p>"X814D78347858EC13" name="X814D78347858EC13" ></a></p>4 .3 -6 IsOne</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsOne</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is the identity morphism.</p>"X7CB5896082D29173" name="X7CB5896082D29173" ></a></p>4 .3 -7 IsIdempotent</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsIdempotent</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is an automorphism.</p>"X78AD1FDD7F53932C" name="X78AD1FDD7F53932C" ></a></p>4 .3 -8 IsMonomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsMonomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is a monomorphism.</p>"X8724CEF182DC4064" name="X8724CEF182DC4064" ></a></p>4 .3 -9 IsEpimorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsEpimorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is an epimorphism.</p>"X7DFACF1F7D7F7EE9" name="X7DFACF1F7D7F7EE9" ></a></p>4 .3 -10 IsSplitMonomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsSplitMonomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is a split monomorphism. <br /></p>"X80A66EFA862E56BC" name="X80A66EFA862E56BC" ></a></p>4 .3 -11 IsSplitEpimorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsSplitEpimorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is a split epimorphism. <br /></p>"X7E07BBF57B92BA56" name="X7E07BBF57B92BA56" ></a></p>4 .3 -12 IsIsomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsIsomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is an isomorphism.</p>"X7F30E3D37E9D7F37" name="X7F30E3D37E9D7F37" ></a></p>4 .3 -13 IsAutomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; IsAutomorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( property )</td ></tr ></table ></div >code class="code" >true</code > or <code class="code" >false</code ></p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > is an automorphism.</p>"X806EEA4685A4A3F3" name="X806EEA4685A4A3F3" ></a></p>4 .4 <span class="Heading" >Morphisms: Attributes</span ></h4>"X7DE8173F80E07AB1" name="X7DE8173F80E07AB1" ></a></p>4 .4 -1 Source </h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; Source </code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > object </p>source of the <strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var >.</p>"X829F76BB80BD55DB" name="X829F76BB80BD55DB" ></a></p>4 .4 -2 Range</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; Range</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > object </p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var >.</p>"X7F3927E287087B64" name="X7F3927E287087B64" ></a></p>4 .4 -3 CokernelEpi</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; CokernelEpi</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>code class="code" >Range</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span > onto the <code class="code" >Cokernel</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span >.</p>"X7D71AE8E838712D7" name="X7D71AE8E838712D7" ></a></p>4 .4 -4 CokernelNaturalGeneralizedIsomorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; CokernelNaturalGeneralizedIsomorphism</code >( <var class="Arg" var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>code class="code" >Cokernel</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span > onto the <code class="code" >Range</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span >.</p>"X87C00FFB79FA93A8" name="X87C00FFB79FA93A8" ></a></p>4 .4 -5 KernelSubobject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; KernelSubobject</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > subobject</p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > as a subobject of the <strong class="pkg" >homalg</strong > object <code class="code" >Source </code >(<var class="Arg" >phi</var >) with generators given by the syzygies of <var class="Arg" >phi</var >.</p>"X82672DB279FAEFCC" name="X82672DB279FAEFCC" ></a></p>4 .4 -6 KernelEmb</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; KernelEmb</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>code class="code" >Kernel</code ><span class="SimpleMath" >(</span ><var "Arg" >phi</var ><span class="SimpleMath" >)</span > into the <code class="code" >Source </code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span >.</p>"X82FB6A4687E778D5" name="X82FB6A4687E778D5" ></a></p>4 .4 -7 ImageSubobject</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ImageSubobject</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > subobject</p>strong class="pkg" >homalg</strong > morphism <var class="Arg" >phi</var > as a subobject of the <strong class="pkg" >homalg</strong > object <code class="code" >Range</code >(<var class="Arg" >phi</var >) with generators given by <var class="Arg" >phi</var > applied to the generators of its source object .</p>"X85FA7C19800F72B2" name="X85FA7C19800F72B2" ></a></p>4 .4 -8 ImageObjectEmb</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ImageObjectEmb</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>code class="code" >ImageObject</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span > into the <code class="code" >Range</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span >.</p>"X86E3E1BA7BCE4D66" name="X86E3E1BA7BCE4D66" ></a></p>4 .4 -9 ImageObjectEpi</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ImageObjectEpi</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>code class="code" >Source </code ><span class="SimpleMath" >(</span var class="Arg" >phi</var ><span class="SimpleMath" >)</span > onto the <code class="code" >ImageObject</code ><span class="SimpleMath" >(</span ><var class="Arg" >phi</var ><span class="SimpleMath" >)</span >.</p>"X823682157C6B4D63" name="X823682157C6B4D63" ></a></p>4 .4 -10 MorphismAid</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; MorphismAid</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>map of a true generalized map . <br /> (no method installed)</p>"X85F22F4084EA7D31" name="X85F22F4084EA7D31" ></a></p>4 .4 -11 InverseOfGeneralizedMorphismWithFullDomain</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; InverseOfGeneralizedMorphismWithFullDomain</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > morphism</p>var class="Arg" >phi</var > (cf. <a href="chapBib.html#biBBaSF" >[Bar09, Cor. 4 .8 ]</a>)).</p>"X8500C49A784C8EDC" name="X8500C49A784C8EDC" ></a></p>4 .4 -12 DegreeOfMorphism</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; DegreeOfMorphism</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( attribute )</td ></tr ></table ></div >var class="Arg" >phi</var > between graded objects. <br /> (no method installed)</p>"X789623548056F7B7" name="X789623548056F7B7" ></a></p>4 .5 <span class="Heading" >Morphisms: Operations and Functions</span ></h4>"X7B4F9EF27A241520" name="X7B4F9EF27A241520" ></a></p>4 .5 -1 ByASmallerPresentation</h5>div class="func" ><table class="func" width="100%" ><tr ><td class="tdleft" ><code class="func" >&#an style='color: green'>8227; ByASmallerPresentation</code >( <var class="Arg" >phi</var > )</td ><td class="tdright" >( method )</td ></tr ></table ></div >strong class="pkg" >homalg</strong > map </p>code class="code" >ByASmallerPresentation</code > for <strong class="pkg" >homalg</strong div class="example" ><pre >"for homalg morphisms" ,Source ( phi ) );pre ></div >var class="Arg" >phi</var > and returns it.</p>div class="example" ><pre >span class="GAPprompt" >gap></span > <span class="GAPinput" >zz := HomalgRingOfIntegers( );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >M := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7 ]" , 2 , 3 , zz );</span >2 x 3 matrix over an internal ring>span class="GAPprompt" >gap></span > <span class="GAPinput" >M := LeftPresentation( M );</span >2 relations for 3 generators>span class="GAPprompt" >gap></span > <span class="GAPinput" >N := HomalgMatrix( "[ 2, 3, 4, 5, 6, 7, 8, 9 ]" , 2 , 4 , zz );</span >2 x 4 matrix over an internal ring>span class="GAPprompt" >gap></span > <span class="GAPinput" >N := LeftPresentation( N );</span >2 relations for 4 generators>span class="GAPprompt" >gap></span > <span class="GAPinput" >mat := HomalgMatrix( "[ \</span> span class="GAPprompt" >></span > <span class="GAPinput" >1 , 0 , -2 , -4 , \</span >span class="GAPprompt" >></span > <span class="GAPinput" >0 , 1 , 4 , 7 , \</span >span class="GAPprompt" >></span > <span class="GAPinput" >1 , 0 , -2 , -4 \</span >span class="GAPprompt" >></span > <span class="GAPinput" >]", 3, 4, zz );</span> 3 x 4 matrix over an internal ring>span class="GAPprompt" >gap></span > <span class="GAPinput" >phi := HomalgMap( mat, M, N );</span >"homomorphism" of left modules>span class="GAPprompt" >gap></span > <span class="GAPinput" >IsMorphism( phi );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >phi;</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( phi );</span >1 , 0 , -2 , -4 ],0 , 1 , 4 , 7 ],1 , 0 , -2 , -4 ] ]map is currently represented by the above 3 x 4 matrixspan class="GAPprompt" >gap></span > <span class="GAPinput" >ByASmallerPresentation( phi );</span >span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( phi );</span >0 , 0 , 0 ],1 , -1 , -2 ] ]map is currently represented by the above 2 x 3 matrixspan class="GAPprompt" >gap></span > <span class="GAPinput" >M;</span >1 left module presented by 1 relation for 2 generators>span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( M );</span >3 > + Z^(1 x 1 )span class="GAPprompt" >gap></span > <span class="GAPinput" >N;</span >2 left module presented by 1 relation for 3 generators>span class="GAPprompt" >gap></span > <span class="GAPinput" >Display( N );</span >4 > + Z^(1 x 2 )pre ></div >div class="chlinkprevnextbot" > <a href="chap0.html" >[Top of Book]</a> <a href="chap0.html#contents" >[Contents]</a> <a href="chap3.html" >[Previous Chapter]</a> <a href="chap5.html" >[Next Chapter]</a> </div >div class="chlinkbot" ><span class="chlink1" >Goto Chapter: </span ><a href="chap0.html" >Top</a> <a href="chap1.html" >1 </a> <a href="chap2.html" >2 </a> <a href="chap3.html" >3 </a> <a href="chap4.html" >4 </a> <a href="chap5.html" >5 </a> <a href="chap6.html" >6 </a> <a href="chap7.html" >7 </a> <a href="chap8.html" >8 </a> <a href="chap9.html" >9 </a> <a href="chap10.html" >10 </a> <a href="chap11.html" >11 </a> <a href="chap12.html" >12 </a> <a href="chapA.html" >A</a> <a href="chapB.html" >B</a> <a href="chapC.html" >C</a> <a href="chapD.html" >D</a> <a href="chapE.html" >E</a> <a href="chapF.html" >F</a> <a href="chapBib.html" >Bib</a> <a href="chapInd.html" >Ind</a> </div >hr />"foot" >generated by <a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc " >GAPDoc2HTML</a></p>body >html > Messung V0.5 in Prozent C=100 H=100 G=100
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