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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (XModAlg) - Chapter 3: Cat1-algebras</title> <meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <meta name="generator" content="GAPDoc2HTML" /> <link rel="stylesheet" type="text/css" href="manual.css" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap3" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <p id="mathjaxlink" class="pcenter"><a href="chap3_mj.html">[MathJax on]</a></p> <p><a id="X85527BA8786CB7FC" name="X85527BA8786CB7FC"></a></p> <div class="ChapSects"><a href="chap3.html#X85527BA8786CB7FC">3 <span class="Heading">Cat1 <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X8 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7B761CD9812972F6">3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7F9561168414C58F">3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X82EC94BA7E7F8DEA">3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X86E99B197E920C21">3 </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3.html#X7 </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X860E29147DA143B5">3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X7A218A0C7DBA8B63">3 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap3.html#X78AE603C857E4EBD">3 </div></div> </div> <h3>3 <span class="Heading">Cat1-algebras</span></h3> <p><a id="X811B6B0F8203F972" name="X811B6B0F8203F972"></a></p> <h4>3.1 <span class="Heading">Definitions and examples</span></h4> <p>Algebraic structures which are equivalent to crossed modules of algebras include :</p> <ul> <li><p>cat<span class="SimpleMath">^1</span>-algebras, (Ellis, <a href="chapBib.html#biBellis </li> <li><p>simplicial algebras with Moore complex of length 1, (Z. Arvasi and T.Porter, <a href="chapB </li> <li><p>algebra-algebroids, (Gaffar Musa's Ph.D. thesis, [Mos86]). </li> </ul> <p>In this section we describe an implementation of cat<span class="SimpleMath">^1</span>-algeb <p>The notion of a cat<span class="SimpleMath">^1</span>-group was defined as an algebraic model o <p>Let <span class="SimpleMath">A</span> and <span class="SimpleMath">R</span> be <span class="Simp <p class="pcenter"> \xymatrix@R=50pt@C=50pt{ A \ar@{->}@<-1.5pt>[d]_{t} \ar@{->}@<1.5pt>[d]^{h} \\ R \ar@/^1.5pc/[u]^{e} } </p> <p>If the conditions,</p> <p class="pcenter"> \mathbf{Cat1Alg1:} \quad te = id_{R} = he, \qquad \mathbf{Cat1Alg2:} \quad (\ker t)(\ker h) = \{0_{A}\} </p> <p>are satisfied, then the algebraic system <span class="SimpleMath">mathcalC := (e;t,h : A → R)</s <p><a id="X7B761CD9812972F6" name="X7B761CD9812972F6"></a></p> <h5>3.1-1 Cat1Algebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>The operations <code class="code">PreCat1AlgebraByEndomorphisms</code> and <code class="co <p><a id="X7F9561168414C58F" name="X7F9561168414C58F"></a></p> <h5>3.1-2 Source</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ta <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ He <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ke <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ke <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Si <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Di <p>These are the nine main attributes of a pre-cat<span class="SimpleMath">^1</span>-algebra.</p> <p>In the example we use homomorphisms between <code class="code">A2c6</code> and <code class="cod <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">t4 := homAR[8]; </span> [ (Z(2)^0)*(1,6,5,4,3,2) ] -> [ (Z(2)^0)*(7,9,8) ] <span class="GAPprompt">gap></span> <span class="GAPinput">e4 := homRA[8];</span> [ (Z(2)^0)*(7,8,9) ] -> [ (Z(2)^0)*(1,5,3)(2,6,4) ] <span class="GAPprompt">gap></span> <span class="GAPinput">C4 := PreCat1AlgebraByTailHeadEmb [AlgebraWithOne( GF(2), [ (Z(2)^0)*(1,2,3,4,5,6) ] ) -> AlgebraWithOne( GF(2), [ (Z(2)^0)*(7,8,9) ] )] <span class="GAPprompt">gap></span> <span class="GAPinput">IsCat1Algebra( C4 );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( C4 );</span> [ 64, 8 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( C4 );</span> [ 6, 3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C4 );</span> Cat1-algebra [..=>..] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3,4,5,6) ] : range algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*(7,8,9) ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*(7,8,9) ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*(7,8,9) ] : range embedding maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*(1,5,3)(2,6,4) ] : kernel has generators: [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6), (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)* (1,5,3)(2,6,4), (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,6,5,4,3,2) ] : boundary homomorphism maps generators of kernel to: [ <zero> of ..., <zero> of ..., <zero> of ... ] : kernel embedding maps generators of kernel to: [ (Z(2)^0)*()+(Z(2)^0)*(1,4)(2,5)(3,6), (Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)* (1,5,3)(2,6,4), (Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,6,5,4,3,2) ] </pre></div> <p><a id="X82EC94BA7E7F8DEA" name="X82EC94BA7E7F8DEA"></a></p> <h5>3.1-3 Cat1AlgebraSelect</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <p>The <code class="func">Cat1Algebra</code> (<a href="chap3.html#X7B761CD9812972F6"><span cla <p>The function <code class="code">Cat1AlgebraSelect</code> may be used in four ways:</p> <ul> <li><p><code class="code">Cat1AlgebraSelect( n )</code> returns the list of possible group orders w </li> <li><p><code class="code">Cat1AlgebraSelect( n, m )</code> returns the list of available group numb </li> <li><p><code class="code">Cat1AlgebraSelect( n, m, k )</code> prints the list of possible cat<span cl </li> <li><p><code class="code">Cat1AlgebraSelect( n, m, k, j )</code> (or simply <code class="code">Cat1A </li> </ul> <p>Now, we give examples of the use of this function.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C := Cat1AlgebraSelect( 11 );</span> |--------------------------------------------------------| | 11 is invalid value for the Galois Field (GFnum) | | Available values for GFnum in the data : | |--------------------------------------------------------| [ 2, 3, 4, 5, 7 ] Usage: Cat1Algebra( GFnum, gpsize, gpnum, num ); fail <span class="GAPprompt">gap></span> <span class="GAPinput">C := Cat1AlgebraSelect( 4, 12 );</spa |--------------------------------------------------------| | 12 is invalid value for size of group (gpsize) | | Available values for gpsize with GF(4) in the data: | |--------------------------------------------------------| Usage: Cat1Algebra( GFnum, gpsize, gpnum, num ); [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ] <span class="GAPprompt">gap></span> <span class="GAPinput">C := Cat1AlgebraSelect( 2, 6, 3 );</sp |--------------------------------------------------------| | 3 is invalid value for groups of order 6 | | Available values for gpnum for groups of size 6 : | |--------------------------------------------------------| Usage: Cat1Algebra( GFnum, gpsize, gpnum, num ); [ 1, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">C := Cat1AlgebraSelect( 2, 6, 2 );</sp There are 4 cat1-structures for the group algebra GF(2)_c6. Range Alg Tail Head |--------------------------------------------------------| | GF(2)_c6 identity map identity map | | ----- [ 2, 10 ] [ 2, 10 ] | | ----- [ 2, 14 ] [ 2, 14 ] | | ----- [ 2, 50 ] [ 2, 50 ] | |--------------------------------------------------------| Usage: Cat1Algebra( GFnum, gpsize, gpnum, num ); Algebra has generators [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3)(4,5) ] 4 </pre></div> <p>The algebra <code class="code">GF(n)_gp</code> has a list of <span class="SimpleMath">n^|gp|</s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C0 := Cat1AlgebraSelect( 4, 6, 2, 2 );< [GF(2^2)_c6 -> Algebra( GF(2^2), [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5)(3,6)+( Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] )] <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( C0 ); </span> [ 4096, 1024 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( C0 );</span> [ 6, 5 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C0 ); </span> Cat1-algebra [GF(2^2)_c6=>..] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*(1,2,3,4,5,6) ] : range algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5) (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5) (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5) (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] : range embedding maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4)(2,5) (3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] : kernel has generators: [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4) (2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] : boundary homomorphism maps generators of kernel to: [ <zero> of ... ] : kernel embedding maps generators of kernel to: [ (Z(2)^0)*()+(Z(2)^0)*(1,2,3,4,5,6)+(Z(2)^0)*(1,3,5)(2,4,6)+(Z(2)^0)*(1,4) (2,5)(3,6)+(Z(2)^0)*(1,5,3)(2,6,4)+(Z(2)^0)*(1,6,5,4,3,2) ] </pre></div> <p><a id="X86E99B197E920C21" name="X86E99B197E920C21"></a></p> <h5>3.1-4 SubCat1Algebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Let <span class="SimpleMath">mathcalC = (e;t,h:A→ R)</span> be a cat<span class="SimpleMath">^1< <p class="pcenter"> t^{\prime} = t|_{A^{\prime}} : A^{\prime}\rightarrow R^{\prime}, \qquad h^{\prime} = h|_{A^{\prime}} : A^{\prime}\rightarrow R^{\prime}, \qquad e^{\prime} = e|_{R^{\prime}} : R^{\prime}\rightarrow A^{\prime} </p> <p>satisfy the <span class="SimpleMath">mathbfCat1Alg1</span> and <span class="SimpleMath">mat <p>If the morphisms satisfy only the <span class="SimpleMath">mathbfCat1Alg1</span> condition t <p>The operations in this subsection are used for constructing subcat<span class="SimpleMath"> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C6 := Cat1AlgebraSelect( 2, 6, 2, 4 ); <span class="GAPprompt">gap></span> <span class="GAPinput">A6 := Source( C6 );</span> GF(2)_c6 <span class="GAPprompt">gap></span> <span class="GAPinput">B6 := Range( C6 );</span> <algebra of dimension 3 over GF(2)> <span class="GAPprompt">gap></span> <span class="GAPinput">eA6 := Elements( A6 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">eB6 := Elements( B6 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SA6 := Subalgebra( A6, [ eA6[1], eA6[ <algebra over GF(2), with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">[ Size(A6), Size(SA6) ]; </span> [ 64, 4 ] <span class="GAPprompt">gap></span> <span class="GAPinput">SB6 := Subalgebra( B6, [ eB6[1], eB6[ <algebra over GF(2), with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">[ Size(B6), Size(SB6) ]; </span> [ 8, 2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">SC6 := SubCat1Algebra( C6, SA6, SB6 ) [Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5) ] ) -> Algebra( GF(2), [ <zero> of ..., (Z(2)^0)*() ] )] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( SC6 );</span> Cat1-algebra [..=>..] :- : source algebra has generators: [ <zero> of ..., (Z(2)^0)*(), (Z(2)^0)*()+(Z(2)^0)*(4,5) ] : range algebra has generators: [ <zero> of ..., (Z(2)^0)*() ] : tail homomorphism maps source generators to: [ <zero> of ..., (Z(2)^0)*(), <zero> of ... ] : head homomorphism maps source generators to: [ <zero> of ..., (Z(2)^0)*(), <zero> of ... ] : range embedding maps range generators to: [ <zero> of ..., (Z(2)^0)*() ] : kernel has generators: [ <zero> of ..., (Z(2)^0)*()+(Z(2)^0)*(4,5) ] : boundary homomorphism maps generators of kernel to: [ <zero> of ..., <zero> of ... ] : kernel embedding maps generators of kernel to: [ <zero> of ..., (Z(2)^0)*()+(Z(2)^0)*(4,5) ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubCat1Algebra( C6, SC6 );</span> true </pre></div> <p><a id="X7DA775CA8296A7D8" name="X7DA775CA8296A7D8"></a></p> <h4>3.2 <span class="Heading">Cat<span class="SimpleMath">^1-</span>algebra morphisms</span></h <p>Let <span class="SimpleMath">mathcalC = (e;t,h:A→ R)</span>, <span class="SimpleMath">mathcal <p class="pcenter"> \xymatrix@R=50pt@C=50pt{ A \ar@{->}@<-1.5pt>[d]_{t} \ar@{->}@<1.5pt>[d]^{h} \ar@{->}[r]^{\phi} & A' \ar@{->}@<-1.5pt>[d]_{t'} \ar@{->}@<1.5pt>[d]^{h'} \\ R \ar@/^1.5pc/[u]^{e} \ar@{->}[r]_{\varphi} & R' \ar@/_1.5 } </p> <p>commutes, (i.e. <span class="SimpleMath">t^' ∘ ϕ = φ ∘ t, h^' ∘ ϕ = φ ∘ h</span> and <span class="Si <p><a id="X860E29147DA143B5" name="X860E29147DA143B5"></a></p> <h5>3.2-1 Cat1AlgebraMorphism</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Id <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ca <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>These operations are used for constructing cat<span class="SimpleMath">^1</span>-algebra mo <p><a id="X7A218A0C7DBA8B63" name="X7A218A0C7DBA8B63"></a></p> <h5>3.2-2 Source</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Na <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <p>These are the six main attributes of a cat<span class="SimpleMath">^1</span>-algebra morphism <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">C1 := Cat1AlgebraSelect( 2, 1, 1, 1 );< [GF(2)_triv -> GF(2)_triv] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C1 );</span> Cat1-algebra [GF(2)_triv=>GF(2)_triv] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*() ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range embedding maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : the kernel is trivial. <span class="GAPprompt">gap></span> <span class="GAPinput">C2 := Cat1AlgebraSelect( 2, 2, 1, 2 );< [GF(2)_c2 -> GF(2)_triv] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( C2 ); </span> Cat1-algebra [GF(2)_c2=>GF(2)_triv] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*(1,2) ] : range algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*() ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range embedding maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : kernel has generators: [ (Z(2)^0)*()+(Z(2)^0)*(1,2) ] : boundary homomorphism maps generators of kernel to: [ <zero> of ... ] : kernel embedding maps generators of kernel to: [ (Z(2)^0)*()+(Z(2)^0)*(1,2) ] <span class="GAPprompt">gap></span> <span class="GAPinput">C1 = C2;</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">SC1 := Source( C1 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SC2 := Source( C2 );</span> GF(2)_c2 <span class="GAPprompt">gap></span> <span class="GAPinput">RC1 := Range( C1 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">RC2 := Range( C2 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">gSC1 := GeneratorsOfAlgebra( SC1 );< [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">gSC2 := GeneratorsOfAlgebra( SC2 );< [ (Z(2)^0)*(), (Z(2)^0)*(1,2) ] <span class="GAPprompt">gap></span> <span class="GAPinput">gRC1 := GeneratorsOfAlgebra( RC1 );< [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">gRC2 := GeneratorsOfAlgebra( RC2 );< [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">imS := [ gSC2[1], gSC2[1] ];</span> [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">homS := AlgebraHomomorphismByImag [ (Z(2)^0)*(), (Z(2)^0)*() ] -> [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">imR := [ gRC2[1], gRC2[1] ];</span> [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">homR := AlgebraHomomorphismByImag [ (Z(2)^0)*(), (Z(2)^0)*() ] -> [ (Z(2)^0)*(), (Z(2)^0)*() ] <span class="GAPprompt">gap></span> <span class="GAPinput">m12 := Cat1AlgebraMorphism( C1, C2, [[GF(2)_triv=>GF(2)_triv] => [GF(2)_c2=>GF(2)_triv]] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( m12 );</span> Morphism of cat1-algebras :- : Source = [GF(2)_triv=>GF(2)_triv] with generating sets: [ (Z(2)^0)*(), (Z(2)^0)*() ] [ (Z(2)^0)*(), (Z(2)^0)*() ] : Range = [GF(2)_c2=>GF(2)_triv] with generating sets: [ (Z(2)^0)*(), (Z(2)^0)*(1,2) ] [ (Z(2)^0)*(), (Z(2)^0)*() ] : Source Homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : Range Homomorphism maps range generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] ## gap> Image2dAlgMapping( m12 ); ## [GF(3)_c2^3=>GF(3)_c2^3] <span class="GAPprompt">gap></span> <span class="GAPinput">IsSurjective( m12 );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsInjective( m12 );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsBijective( m12 );</span> false </pre></div> <p><a id="X78AE603C857E4EBD" name="X78AE603C857E4EBD"></a></p> <h5>3.2-3 ImagesSource2DimensionalMapping</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Im <p>When <span class="SimpleMath">(θ,φ)</span> is a homomorphism of cat1-algebras (or crossed modul <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">im12 := ImagesSource2DimensionalM <span class="GAPprompt">gap></span> <span class="GAPinput">Display( im12 ); </span> Cat1-algebra [..=>..] :- : source algebra has generators: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range algebra has generators: [ (Z(2)^0)*() ] : tail homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : head homomorphism maps source generators to: [ (Z(2)^0)*(), (Z(2)^0)*() ] : range embedding maps range generators to: [ (Z(2)^0)*() ] : the kernel is trivial. </pre></div> <div class="chlinkprevnextbot"> <a href="chap0.html">[Top of Book]</a> <a href="chap0.html#con <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA 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2026-03-28