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<span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X79A3A0CF82B6F08 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X86E1182E7EEFAAD <span class="ContSS"><br /><span class="nocss"> </span><a href="chap7_mj.html#X7FDAEAFF78A5E7D </div></div> </div> <h3>7 <span class="Heading">Self-similar groups, monoids and semigroups</span></h3> <p>Self-similar groups, monoids and semigroups (below <em>FR semigroups</em>) are simply groups, <p>Most non-trivial calculations in FR groups are performed as follows: <strong class="pkg">GAP< <p>The maximal length of words to consider in the search is controlled by the variable <code class= <p><a id="X80A26BAA7B53C1BD" name="X80A26BAA7B53C1BD"></a></p> <h4>7.1 <span class="Heading">Creators for FR semigroups</span></h4> <p>The most straightforward creation method for FR groups is <code class="code">Group()</code>, a <p><a id="X7AE8F92383272329" name="X7AE8F92383272329"></a></p> <h5>7.1-1 FRGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <p>Returns: A new self-similar group/monoid/semigroup.</p> <p>This function constructs a new FR group/monoid/semigroup, generated by group FR elements. I <p>Each <var class="Arg">definition</var> is of the form <code class="code">"name=projtrans"</cod <p>The last argument may be one of the filters <code class="code">IsMealyElement</code>, <code clas <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">FRGroup("a=(1,2)","b=(1,2,3,4,5 <self-similar group over [ 1 .. 5 ] with 2 generators> 120 <span class="GAPprompt">gap></span> <span class="GAPinput">Dinfinity := FRGroup("a=(1,2)","b <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Dinfin #I Assigned the global variables [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">Order(a); Order(b); Order(a*b);</s 2 2 infinity <span class="GAPprompt">gap></span> <span class="GAPinput">ZZ := FRGroup("t=<,t>[2,1]");</span> <self-similar group over [ 1 .. 2 ] with 1 generator> tau := FRElement([[[b,1],[1]]],[()],[1]); <2|f3> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(Dinfinity,ZZ);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(Dinfinity^tau,ZZ);</s true <span class="GAPprompt">gap></span> <span class="GAPinput">Index(Dinfinity^tau,ZZ);</span> 2 </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i4 := FRMonoid("s=(1,2)","f=<s,f>[1 <self-similar monoid over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">f := GeneratorsOfMonoid(i4){[1,2] <span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..10] do Add(f,f[i]*f[i+1 <span class="GAPprompt">gap></span> <span class="GAPinput">f[1]^2=One(m);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">f[2]^3=f[2];</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">f[11]*f[10]^2=f[1]*Product(f{[5 true <span class="GAPprompt">gap></span> <span class="GAPinput">f[12]*f[11]^2=f[2]*Product(f{[6 true </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i2 := FRSemigroup("f0=<f0,f0>(1,2)" <self-similar semigroup over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(i2);</s #I Assigned the global variables [ "f0", "f1" ] <span class="GAPprompt">gap></span> <span class="GAPinput">f0^2=One(i2);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([0..10],p->(f0*f1)^p*(f1* true </pre></div> <p><a id="X7D4A6996874A3DF3" name="X7D4A6996874A3DF3"></a></p> <h5>7.1-2 NewSemigroupFRMachine</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ne <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ne <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ne <p>Returns: A new FR machine, based on string descriptions.</p> <p>This command constructs a new FR machine, in a format similar to <code class="func">FRGroup</co <p>Except in the semigroup case, <code class="code">word-i</code> is allowed to be the empty string <p>The following example constructs the "universal Grigorchuk machine".</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">m := NewGroupFRMachine("a=(1,2)(3 "c=<a,c,,c,a,c>","d=<,d,a,d,a,d>"); <span class="GAPprompt">gap></span> <span class="GAPinput"><FR machine with alphabet [ 1, 2, 3, 4, 5 </pre></div> <p><a id="X853E3F0680C76F56" name="X853E3F0680C76F56"></a></p> <h5>7.1-3 SCGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SC <p>Returns: The state-closed group/monoid/semigroup generated by the machine <var class="Arg <p>This function constructs a new FR group/monoid/semigroup <code class="code">g</code>, genera <p>In the non-<code class="code">NC</code> forms, redundant (equal, trivial or mutually inverse) <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">b := MealyMachine([[3,2],[3,1],[3 <self-similar group over [ 1 .. 2 ] with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Size(g);</span> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">IsOne(Comm(g.2,g.2^g.1));</span> true </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i4machine := MealyMachine([[3,3], <Mealy machine on alphabet [ 1, 2 ] with 3 states> <span class="GAPprompt">gap></span> <span class="GAPinput">IsInvertible(i4machine);</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">i4 := SCMonoidNC(i4machine);</span> <self-similar monoid over [ 1 .. 2 ] with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">f := GeneratorsOfMonoid(i4){[1,2] <span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..10] do Add(f,f[i]*f[i+1 <span class="GAPprompt">gap></span> <span class="GAPinput">f[1]^2=One(m);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">f[2]^3=f[2];</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">f[11]*f[10]^2=f[1]*Product(f{[5 true <span class="GAPprompt">gap></span> <span class="GAPinput">f[12]*f[11]^2=f[2]*Product(f{[6 true </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i2machine := MealyMachine([[1,1], <Mealy machine on alphabet [ 1, 2 ] with 2 states> <span class="GAPprompt">gap></span> <span class="GAPinput">i2 := SCSemigroupNC(i2machine);</s <self-similar semigroup over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">f0 := GeneratorsOfSemigroup(i2)[1 <span class="GAPprompt">gap></span> <span class="GAPinput">f0^2=One(i2);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([0..10],p->(f0*f1)^p*(f1* true </pre></div> <p><a id="X7F15D57A7959FEF6" name="X7F15D57A7959FEF6"></a></p> <h5>7.1-4 Correspondence</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Co <p>Returns: A correspondence between the generators of the underlying FR machine of <var class=" <p>If <var class="Arg">g</var> was created as the state closure of an FR machine <code class="code">m</ <p>If <code class="code">m</code> is a group/monoid/semigroup/algebra FR machine, then <code clas <p>If <code class="code">m</code> is a Mealy or vector machine, then <code class="code">Corresponde <p>See <code class="func">SCGroupNC</code> (<a href="chap7_mj.html#X853E3F0680C76F56"><span cl <p><a id="X7D0B8334786E2802" name="X7D0B8334786E2802"></a></p> <h5>7.1-5 FullSCGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fu <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Fu <p>Returns: A maximal state-closed group/monoid/semigroup on the alphabet <var class="Arg">a</ <p>This function constructs a new FR group, monoid or semigroup, which contains all transformat <p>The arguments can be, in any order: a semigroup, specifying which vertex actions are allowed; <p>This object serves as a container for all FR elements with alphabet <var class="Arg">a</var>. Ran <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">g := FullSCGroup([1..3]);</span> FullSCGroup([ 1 .. 3 ]); <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(g,GuptaSidkiGroup);</ true <span class="GAPprompt">gap></span> <span class="GAPinput">g := FullSCGroup([1..3],Group((1, FullSCGroup([ 1 .. 3 ], Group( [ (1,2,3) ] )) <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(g,GuptaSidkiGroup);</ true <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubgroup(g,GrigorchukGroup);</ false <span class="GAPprompt">gap></span> <span class="GAPinput">Random(g);</span> <Mealy element on alphabet [ 1, 2, 3 ] with 2 states, initial state 1> <span class="GAPprompt">gap></span> <span class="GAPinput">Size(FullSCGroup([1,2],3));</spa 128 <span class="GAPprompt">gap></span> <span class="GAPinput">g := FullSCMonoid([1..2]);</span> FullSCMonoid([ 1 .. 2 ]) <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(g,AsTransformation(Ful true <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(g,AsTransformation(Gri true <span class="GAPprompt">gap></span> <span class="GAPinput">g := FullSCSemigroup([1..3]);</spa FullSCSemigroup([ 1 .. 3 ]) <span class="GAPprompt">gap></span> <span class="GAPinput">h := FullSCSemigroup([1..3],Semig FullSCSemigroup([ 1 .. 3 ], Semigroup( [ [ 1, 1, 1 ] ] )) <span class="GAPprompt">gap></span> <span class="GAPinput">Size(h);</span> 1 <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubset(g,h);</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">g=FullSCMonoid([1..3]);</span> true </pre></div> <p><a id="X7DB92C34827D513F" name="X7DB92C34827D513F"></a></p> <h5>7.1-6 FRMachineFRGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Me <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Me <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Me <p>Returns: A machine describing all generators of <var class="Arg">g</var>.</p> <p>This function constructs a new group/monoid/semigroup/Mealy FR machine, with (at least) on <p>In particular, if <var class="Arg">g</var> is state-closed, then <code class="code">SCGroup(FR <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">FRMachineFRGroup(GuptaSidkiGrou <FR machine with alphabet [ 1 .. 3 ] on Group( [ f11, f12 ] )> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span> G | 1 2 3 -----+--------+----------+--------+ f11 | <id>,2 <id>,3 <id>,1 f12 | f11,1 f11^-1,2 f12,3 -----+--------+----------+--------+ </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">FRMachineFRMonoid(I4Monoid);</sp <FR machine with alphabet [ 1 .. 2 ] on Monoid( [ m11, m12 ], ... )> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span> M | 1 2 -----+--------+--------+ m11 | <id>,2 <id>,1 m12 | m11,1 m12,1 -----+--------+--------+ </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">FRMachineFRSemigroup(I2Monoid);< <FR machine with alphabet [ 1 .. 2 ] on Semigroup( [ s11, s12, s1 ] )> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(last);</span> S | 1 2 -----+-------+-------+ s11 | s11,1 s11,2 s12 | s12,2 s12,1 s1 | s1,2 s12,2 -----+-------+-------+ </pre></div> <p><a id="X7BF4AC9F830A8E1A" name="X7BF4AC9F830A8E1A"></a></p> <h5>7.1-7 IsomorphismFRGroup</h5> <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>Returns: An isomorphism towards a group/monoid/semigroup on a single FR machine.</p> <p>This function constructs a new FR group/monoid/semigroup, such that all elements of the resu <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismFRGroup(GuptaSi [ <Mealy element on alphabet [ 1, 2, 3 ] with 2 states, initial state 1>, <Mealy element on alphabet [ 1, 2, 3 ] with 4 states, initial state 1> ] -> [ <3|identity ...>, <3|f1>, <3|f1^-1>, <3|f2> ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GuptaSidkiGroup.2);</spa | 1 2 3 ---+-----+-----+-----+ a | a,1 a,2 a,3 b | a,2 a,3 a,1 c | a,3 a,1 a,2 d | b,1 c,2 d,3 ---+-----+-----+-----+ Initial state: d <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GuptaSidkiGroup.2^phi);< | 1 2 3 ----+--------+---------+--------+ f1 | <id>,2 <id>,3 <id>,1 f2 | f1,1 f1^-1,2 f2,3 ----+--------+---------+--------+ Initial state: f2 </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismFRSemigroup(I2M MappingByFunction( I2, <self-similar semigroup over [ 1 .. 2 ] with 3 generators>, <Operation "AsSemigroupFRElement"> ) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GeneratorsOfSemigroup(I | 1 2 ---+-----+-----+ a | a,2 b,2 b | b,2 b,1 ---+-----+-----+ Initial state: a <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GeneratorsOfSemigroup(I S | 1 2 ----+------+------+ s1 | s1,2 s2,2 s2 | s2,2 s2,1 ----+------+------+ Initial state: s1 </pre></div> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismFRMonoid(I4Mono MappingByFunction( I4, <self-similar monoid over [ 1 .. 2 ] with 2 generators>, <Operation "AsMonoidFRElement"> ) <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GeneratorsOfMonoid(I4Mo | 1 2 ---+-----+-----+ a | b,2 b,1 b | b,1 b,2 ---+-----+-----+ Initial state: a <span class="GAPprompt">gap></span> <span class="GAPinput">Display(GeneratorsOfMonoid(I4Mo M | 1 2 ----+--------+--------+ m1 | <id>,2 <id>,1 ----+--------+--------+ Initial state: m1 </pre></div> <p><a id="X7DE1CAE981F2825B" name="X7DE1CAE981F2825B"></a></p> <h5>7.1-8 IsomorphismMealyGroup</h5> <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>Returns: An isomorphism towards a group/monoid/semigroup all of whose elements are Mealy ma <p>This function constructs a new FR group/monoid/semigroup, such that all elements of the resu <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := FRGroup("a=(1,2)","b=<a,b>","c <self-similar group over [ 1 .. 2 ] with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">phi := IsomorphismMealyGroup(G);</ [ <2|a>, <2|b>, <2|c> ] -> [ <Mealy element on alphabet [ 1, 2 ] with 2 states, initial state 1>, <Mealy element on alphabet [ 1, 2 ] with 3 states, initial state 1>, <Mealy element on alphabet [ 1, 2 ] with 4 states, initial state 1> ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display(G.3);</span> | 1 2 ---+--------+--------+ a | <id>,2 <id>,1 b | a,1 b,2 c | c,1 b,2 ---+--------+--------+ Initial state: c <span class="GAPprompt">gap></span> <span class="GAPinput">Display(G.3^phi);</span> | 1 2 ---+-----+-----+ a | a,1 b,2 b | c,1 b,2 c | d,2 d,1 d | d,1 d,2 ---+-----+-----+ Initial state: a </pre></div> <p><a id="X7BB8DDEA83946C73" name="X7BB8DDEA83946C73"></a></p> <h5>7.1-9 FRGroupByVirtualEndomorphism</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FR <p>Returns: A new self-similar group.</p> <p>This function constructs a new FR group <code class="code">P</code>, generated by group FR eleme <p>The optional second argument is a transversal of <code class="code">H</code> in <code class="cod <p>Furthermore, an option "MealyElement" can be passed to the function, as <code class="code">FR <p>The resulting FR group has an attribute <code class="code">Correspondence(P)</code> that reco <p>The example below constructs the binary adding machine, and a non-standard representation o <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := FreeGroup(1);</span> <free group on the generators [ f1 ]> <span class="GAPprompt">gap></span> <span class="GAPinput">f := GroupHomomorphismByImages(Gr [ f1^2 ] -> [ f1 ] <span class="GAPprompt">gap></span> <span class="GAPinput">H := FRGroupByVirtualEndomorphism <self-similar group over [ 1 .. 2 ] with 1 generator> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(H.1);</span> | 1 2 ----+--------+------+ x1 | <id>,2 x1,1 ----+--------+------+ Initial state: x1 <span class="GAPprompt">gap></span> <span class="GAPinput">Correspondence(H);</span> [ f1 ] -> [ <2|x1> ] <span class="GAPprompt">gap></span> <span class="GAPinput">H := FRGroupByVirtualEndomorphism <span class="GAPprompt">gap></span> <span class="GAPinput">Display(H.1);</span> | 1 2 ----+---------+--------+ x1 | x1^-1,2 x1^2,1 ----+---------+--------+ Initial state: x1 <span class="GAPprompt">gap></span> <span class="GAPinput">H := FRGroupByVirtualEndomorphism <self-similar group over [ 1 .. 2 ] with 1 generator> <span class="GAPprompt">gap></span> <span class="GAPinput">Display(H.1);</span> | 1 2 ---+-----+-----+ a | b,2 a,1 b | b,1 b,2 ---+-----+-----+ Initial state: a </pre></div> <p><a id="X79D75A7D80DD9AD1" name="X79D75A7D80DD9AD1"></a></p> <h5>7.1-10 TreeWreathProduct</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <p>Returns: The tree-wreath product of groups <var class="Arg">g,h</var>.</p> <p>The tree-wreath product of two FR groups is a group generated by a copy of <var class="Arg">g</var> <p>More formally, assume without loss of generality that all generators of <var class="Arg">g</va <p>For the operation on FR machines see <code class="func">TreeWreathProduct</code> (<a href="cha <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">w := TreeWreathProduct(AddingGrou <recursive group over [ 1 .. 4 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">a := w.1; b := w.2;</span> <Mealy element on alphabet [ 1 .. 4 ] with 3 states> <Mealy element on alphabet [ 1 .. 4 ] with 2 states> <span class="GAPprompt">gap></span> <span class="GAPinput">Order(a); Order(b);</span> infinity infinity <span class="GAPprompt">gap></span> <span class="GAPinput">ForAll([-100..100],i->IsOne(Comm true </pre></div> <p>the group <code class="code">w</code> is the wreath product <span class="SimpleMath">\(Z\wr Z\)< <p><a id="X85840A047C04BFC6" name="X85840A047C04BFC6"></a></p> <h5>7.1-11 WeaklyBranchedEmbedding</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ We <p>Returns: A embedding of <var class="Arg">g</var> in a weakly branched group.</p> <p>This function constructs a new FR group, on alphabet the square of the alphabet of <var class="A <p>The main result of <a href="chapBib_mj.html#biBMR1995624">[SW03]</a> is that the resulting gr <p><a id="X84E20571841DE1E4" name="X84E20571841DE1E4"></a></p> <h4>7.2 <span class="Heading">Operations for FR semigroups</span></h4> <p><a id="X7C6D7BA0818A3A3D" name="X7C6D7BA0818A3A3D"></a></p> <h5>7.2-1 PermGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pe <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <p>Returns: [An epimorphism to] the permutation group of <var class="Arg">g</var>'s action on leve <p>The first function returns a permutation group on <span class="SimpleMath">\(d^l\)</span> poi <p>The second function returns a homomorphism from <var class="Arg">g</var> to this permutation gr <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">g := FRGroup("a=(1,2)","b=<a,>"); Si <self-similar group over [ 1 .. 2 ] with 2 generators> 8 <span class="GAPprompt">gap></span> <span class="GAPinput">PermGroup(g,2);</span> Group([ (1,3)(2,4), (1,2) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">PermGroup(g,3);</span> Group([ (1,5)(2,6)(3,7)(4,8), (1,3)(2,4) ]) <span class="GAPprompt">gap></span> <span class="GAPinput">List([1..6],i->LogInt(Size(PermG [ 1, 3, 7, 12, 22, 42 ] <span class="GAPprompt">gap></span> <span class="GAPinput">g := FRGroup("t=<,t>(1,2)"); Size(g) <self-similar group over [ 1 .. 2 ] with 1 generator> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">pi := EpimorphismPermGroup(g,5);</ MappingByFunction( <self-similar group over [ 1 .. 2 ] with 1 generator, of size infinity>, Group([ (1,17,9,25,5,21,13,29,3,19,11,27,7,23,15,31, 2,18,10,26,6,22,14,30,4,20,12,28,8,24,16,32) ]), function( w ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">Order(g.1);</span> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">Order(g.1^pi);</span> 32 </pre></div> <p><a id="X8620BEAF7957FA4D" name="X8620BEAF7957FA4D"></a></p> <h5>7.2-2 PcGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pc <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <p>Returns: [An epimorphism to] the pc group of <var class="Arg">g</var>'s action on level <var class <p>The first function returns a polycyclic group representing the action of <var class="Arg">g</v <p>The second function returns a homomorphism from <var class="Arg">g</var> to this pc group.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">g := PcGroup(GrigorchukGroup,7); t <pc group with 5 generators> 3370 <span class="GAPprompt">gap></span> <span class="GAPinput">NormalClosure(g,Group(g.3)); tim <pc group with 79 generators> 240 <span class="GAPprompt">gap></span> <span class="GAPinput">g := PermGroup(GrigorchukGroup,7) <permutation group with 5 generators> 3 <span class="GAPprompt">gap></span> <span class="GAPinput">NormalClosure(g,Group(g.3)); tim <permutation group with 5 generators> 5344 <span class="GAPprompt">gap></span> <span class="GAPinput">g := FRGroup("t=<,t>(1,2)"); Size(g) <self-similar group over [ 1 .. 2 ] with 1 generator> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">pi := EpimorphismPcGroup(g,5);</sp MappingByFunction( <self-similar group over [ 1 .. 2 ] with 1 generator, of size infinity>, Group([ f1, f2, f3, f4, f5 ]), function( w ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">Order(g.1);</span> infinity <span class="GAPprompt">gap></span> <span class="GAPinput">Order(g.1^pi);</span> 32 </pre></div> <p><a id="X83834FF77F972912" name="X83834FF77F972912"></a></p> <h5>7.2-3 TransformationMonoid</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <p>Returns: [An epimorphism to] the transformation monoid of <var class="Arg">g</var>'s action on <p>The first function returns a transformation monoid on <span class="SimpleMath">\(d^l\)</spa <p>The second function returns a homomorphism from <var class="Arg">g</var> to this transformatio <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i4 := SCMonoid(MealyMachine([[3,3 <self-similar monoid over [ 1 .. 2 ] with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">g := TransformationMonoid(i4,6);</ <monoid with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">List([1..6],i->Size(Transformati [ 4, 14, 50, 170, 570, 1882 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Collected(List(g,RankOfTransfor [ [ 1, 64 ], [ 2, 1280 ], [ 4, 384 ], [ 8, 112 ], [ 16, 32 ], [ 32, 8 ], [ 64, 2 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">pi := EpimorphismTransformationMo MappingByFunction( <self-similar monoid over [ 1 .. 2 ] with 3 generators>, <monoid with 3 generators>, function( w ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">f := GeneratorsOfMonoid(i4){[1,2] <span class="GAPprompt">gap></span> <span class="GAPinput">for i in [1..10] do Add(f,f[i]*f[i+1 <span class="GAPprompt">gap></span> <span class="GAPinput">Product(f{[3,5,7,9,11]})=f[11]* false <span class="GAPprompt">gap></span> <span class="GAPinput">Product(f{[3,5,7,9,11]})^pi=(f[ true </pre></div> <p><a id="X8768C22D859BE75F" name="X8768C22D859BE75F"></a></p> <h5>7.2-4 TransformationSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Tr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <p>Returns: [An epimorphism to] the transformation semigroup of <var class="Arg">g</var>'s actio <p>The first function returns a transformation semigroup on <span class="SimpleMath">\(d^l\)</ <p>The second function returns a homomorphism from <var class="Arg">g</var> to this transformatio <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">i2 := SCSemigroup(MealyMachine([[ <self-similar semigroup over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">g := TransformationSemigroup(i2,6 <semigroup with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">List([1..6],i->Size(Transformati [ 4, 14, 42, 114, 290, 706 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Collected(List(g,RankOfTransfor [ [ 1, 64 ], [ 2, 384 ], [ 4, 160 ], [ 8, 64 ], [ 16, 24 ], [ 32, 8 ], [ 64, 2 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">f0 := GeneratorsOfSemigroup(i2)[1 <span class="GAPprompt">gap></span> <span class="GAPinput">pi := EpimorphismTransformationSe MappingByFunction( <self-similar semigroup over [ 1 .. 2 ] with 2 generators>, <semigroup with 2 generators>, function( w ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">(f1*(f1*f0)^10)=((f1*f0)^10);</s false <span class="GAPprompt">gap></span> <span class="GAPinput">(f1*(f1*f0)^10)^pi=((f1*f0)^10) true </pre></div> <p><a id="X7BDC634086437315" name="X7BDC634086437315"></a></p> <h5>7.2-5 EpimorphismGermGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ep <p>Returns: A homomorphism to a polycyclic group.</p> <p>This function returns an epimorphism to a polycyclic group, encoding the action on the first <v <p>Since the elements of <var class="Arg">g</var> are finite automata, they map periodic sequences <p>The germ group, by default, is abelian. If it is finite, this function returns a homomorphism t <p>The <code class="func">GrigorchukTwistedTwin</code> (<a href="chap9_mj.html#X7E765AF77AA <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismGermGroup(Grigorchuk MappingByFunction( GrigorchukGroup, <pc group of size 4 with 2 generators>, function( g ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">List(GeneratorsOfGroup(Grigorch [ <identity> of ..., f1, f1*f2, f2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">StructureDescription(Image(last "C2 x C2" <span class="GAPprompt">gap></span> <span class="GAPinput">g := FRGroup("t=<,t>(1,2)","m=<,m^-1> <span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismGermGroup(g,0);</span> MappingByFunction( <state-closed, bounded group over [ 1, 2 ] with 2 generators>, Pcp-group with orders [ 0, 0 ], function( x ) ... end ) <span class="GAPprompt">gap></span> <span class="GAPinput">EpimorphismGermGroup(g,1);; Rang Pcp-group with orders [ 2, 0, 0, 0, 0 ] Pcp-group with orders [ 2, 0, 0, 0 ] </pre></div> <p><a id="X812242E584462766" name="X812242E584462766"></a></p> <h5>7.2-6 GermData</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ge <p>The first command computes some data useful to determine the germ value of a group element; the <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">data := GermData(GrigorchukGroup) rec( endo := [ f1, f2 ] -> [ f1*f2, f1 ], group := <pc group of size 4 with 2 generators>, machines := [ ], map := [ <identity> of ..., f2, f1, f1*f2, <identity> of ... ], nucleus := [ <Trivial Mealy element on alphabet [ 1 .. 2 ]>, d, c, b, a ], nucleusmachine := <Mealy machine on alphabet [ 1 .. 2 ] with 5 states> ) <span class="GAPprompt">gap></span> <span class="GAPinput">List(GeneratorsOfGroup(Grigorch [ <identity> of ..., f1*f2, f1, f2 ] </pre></div> <p><a id="X87378D53791D0B70" name="X87378D53791D0B70"></a></p> <h5>7.2-7 StabilizerImage</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ St <p>Returns: The group of all states at <var class="Arg">v</var> of elements of <var class="Arg">g</var> <p>This function constructs a new FR group, consisting of all states at vertex <var class="Arg">v</ <p>The result is <var class="Arg">g</var> itself precisely if <var class="Arg">g</var> is recurrent (s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := FRGroup("t=<,t>(1,2)","u=<,u^-1> <self-similar group over [ 1 .. 2 ] with 3 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Stabilizer(G,1);</span> <self-similar group over [ 1 .. 2 ] with 5 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(last);</span> [ <2|u*t^-1>, <2|b>, <2|t^2>, <2|t*u>, <2|t*b*t^-1> ] <span class="GAPprompt">gap></span> <span class="GAPinput">StabilizerImage(G,1);</span> <self-similar group over [ 1 .. 2 ] with 5 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfGroup(last);</span> [ <2|identity ...>, <2|u>, <2|t>, <2|u^-1>, <2|t> ] </pre></div> <p><a id="X7B4CD9CA872BA368" name="X7B4CD9CA872BA368"></a></p> <h5>7.2-8 LevelStabilizer</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Le <p>Returns: The fixator of the <var class="Arg">n</var>th level of the tree.</p> <p>This function constructs the normal subgroup of <var class="Arg">g</var> that fixes the <var clas <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">G := FRGroup("t=<,t>(1,2)","a=(1,2) <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">LevelStabilizer(G,2);</span> <self-similar group over [ 1 .. 2 ] with 9 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Index(G,last);</span> 8 <span class="GAPprompt">gap></span> <span class="GAPinput">IsNormal(G,last2);</span> true </pre></div> <p><a id="X7C5002E683A044C1" name="X7C5002E683A044C1"></a></p> <h5>7.2-9 IsStateClosed</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="keyw">true</code> if all states of elements of <var class="Arg">g</var> belo <p>This function tests whether <var class="Arg">g</var> is a <em>state-closed</em> group, i.e. a group <p>The smallest state-closed group containing <var class="Arg">g</var> is computed with <code clas <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Dinfinity := FRGroup("a=(1,2)","b <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Dinfin #I Assigned the global variables [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsStateClosed(Group(a));</span> IsStateClosed(Group(b)); IsStateClosed(Dinfinity); true false true </pre></div> <p><a id="X79246DB482BEAF2D" name="X79246DB482BEAF2D"></a></p> <h5>7.2-10 StateClosure</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ St <p>Returns: The smallest state-closed group containing <var class="Arg">g</var>.</p> <p>This function computes the smallest group containing all states of all elements of <var class= <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Dinfinity := FRGroup("a=(1,2)","b <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Dinfin #I Assigned the global variables [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">StateStateClosure(Group(a))=Din false true </pre></div> <p><a id="X7E2F34417EBB7673" name="X7E2F34417EBB7673"></a></p> <h5>7.2-11 IsRecurrentFRSemigroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="keyw">true</code> if <var class="Arg">g</var> is a recurrent group.</p> <p>This function returns <code class="keyw">true</code> if <var class="Arg">g</var> is a <em>recurrent< <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Dinfinity := FRGroup("a=(1,2)","b <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Dinfin #I Assigned the global variables [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsRecurrentFRSemigroup(Group(a) false false <span class="GAPprompt">gap></span> <span class="GAPinput">IsRecurrentFRSemigroup(Dinfinit true </pre></div> <p><a id="X825688FD7CE96479" name="X825688FD7CE96479"></a></p> <h5>7.2-12 IsLevelTransitiveFRGroup</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Returns: <code class="keyw">true</code> if <var class="Arg">g</var> is a level-transitive group.< <p>This function returns <code class="keyw">true</code> if <var class="Arg">g</var> is a <em>level-tra <p>This function can be abbreviated as <code class="code">IsLevelTransitive</code>.</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Dinfinity := FRGroup("a=(1,2)","b <self-similar group over [ 1 .. 2 ] with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">AssignGeneratorVariables(Dinfin #I Assigned the global variables [ a, b ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsLevelTransitiveFRGroup(Group( IsLevelTransitiveFRGroup(Dinfinity); false false true </pre></div> <p><a id="X7D95219481AEDD20" name="X7D95219481AEDD20"></a></p> <h5>7.2-13 IsInfinitelyTransitive</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is |
