<!-- #################################################################### --> <!-- ## ## --> <!-- ## rcwamono.xml RCWA documentation Stefan Kohl ## --> <!-- ## ## --> <!-- ## $Id: rcwamono.xml,v 1.12 2007/09/26 13:20:16 stefan Exp $ ## --> <!-- ## ## --> <!-- #################################################################### --> <Chapter Label="ch:RcwaMonoids"> <Heading>Residue-Class-Wise Affine Monoids</Heading> <Ignore Remark="set screen width to 75, for the example tester"> <Example> <![CDATA[ gap> SizeScreen([75,24]);; ]]> </Example> </Ignore> In this short chapter, we describe how to compute with residue-class-wise affine monoids. <Index Key="rcwa monoid" Subkey="definition">rcwa monoid</Index> <E>Residue-class-wise affine</E> monoids, or <E>rcwa</E> monoids for short, are monoids whose elements are residue-class-wise affine mappings. <!-- #################################################################### --> <Section Label="sec:ContructingRcwaMonoids"> <Heading>Constructing residue-class-wise affine monoids</Heading> <Index Key="Monoid"><C>Monoid</C></Index> <Index Key="MonoidByGenerators"><C>MonoidByGenerators</C></Index> As any other monoids in &GAP;, residue-class-wise affine monoids can be constructed by <C>Monoid</C> or <C>MonoidByGenerators</C>. <Example> <![CDATA[ gap> M := Monoid(RcwaMapping([[ 0,1,1],[1,1,1]]), > RcwaMapping([[-1,3,1],[0,2,1]])); <rcwa monoid over Z with 2 generators> gap> Size(M); 11 gap> Display(MultiplicationTable(M)); [ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [ 2, 8, 5, 11, 8, 3, 10, 5, 2, 8, 5 ], [ 3, 10, 11, 5, 5, 5, 8, 8, 8, 2, 3 ], [ 4, 9, 6, 8, 8, 8, 5, 5, 5, 7, 4 ], [ 5, 8, 5, 8, 8, 8, 5, 5, 5, 8, 5 ], [ 6, 7, 4, 8, 8, 8, 5, 5, 5, 9, 6 ], [ 7, 5, 8, 6, 5, 4, 9, 8, 7, 5, 8 ], [ 8, 5, 8, 5, 5, 5, 8, 8, 8, 5, 8 ], [ 9, 5, 8, 4, 5, 6, 7, 8, 9, 5, 8 ], [ 10, 8, 5, 3, 8, 11, 2, 5, 10, 8, 5 ], [ 11, 2, 3, 5, 5, 5, 8, 8, 8, 10, 11 ] ] ]]> </Example> <Index Key="View" Subkey="for an rcwa monoid"><C>View</C></Index> <Index Key="Display" Subkey="for an rcwa monoid"><C>Display</C></Index> <Index Key="Print" Subkey="for an rcwa monoid"><C>Print</C></Index> <Index Key="String" Subkey="for an rcwa monoid"><C>String</C></Index> There are methods for the operations <C>View</C>, <C>Display</C>, <C>Print</C> and <C>String</C> which are applicable to rcwa monoids. All rcwa monoids over a ring <M>R</M> are submonoids of Rcwa(<M>R</M>). The monoid Rcwa(<M>R</M>) itself is not finitely generated, thus cannot be constructed as described above. It is handled as a special case: <ManSection> <Func Name="Rcwa" Arg="R" Label="the monoid of all rcwa mappings of a ring"/> <Returns> The monoid Rcwa(<A>R</A>) of all residue-class-wise affine mappings of the ring <A>R</A>. </Returns> <Description> <Example> <![CDATA[ gap> RcwaZ := Rcwa(Integers); Rcwa(Z) gap> IsSubset(RcwaZ,M); true ]]> </Example> </Description> </ManSection> <Index Key="Restriction" Subkey="for an rcwa monoid, by an injective rcwa mapping"> <C>Restriction</C> </Index> <Index Key="Induction" Subkey="for an rcwa monoid, by an injective rcwa mapping"> <C>Induction</C> </Index> In our methods to construct rcwa groups, two kinds of mappings played a crucial role, namely the restriction monomorphisms (cf. <Ref Oper="Restriction" Label="of an rcwa group, by an injective rcwa mapping"/>) and the induction epimorphisms (cf. <Ref Oper="Induction" Label="of an rcwa group, by an injective rcwa mapping"/>). The restriction monomorphisms extend in a natural way to the monoids Rcwa(<M>R</M>), and the induction epimorphisms have corresponding generalizations, also. Therefore the operations <C>Restriction</C> and <C>Induction</C> can be applied to rcwa monoids as well: <Example> <![CDATA[ gap> M2 := Restriction(M,2*One(Rcwa(Integers))); <rcwa monoid over Z with 2 generators, of size 11> gap> Support(M2); 0(2) gap> Action(M2,ResidueClass(1,2)); Trivial rcwa group over Z gap> Induction(M2,2*One(Rcwa(Integers))) = M; true ]]> </Example> </Section> <!-- #################################################################### --> <Section Label="sec:ComputingWithRcwaMonoids"> <Heading>Computing with residue-class-wise affine monoids</Heading> <Index Key="Size" Subkey="for an rcwa monoid"><C>Size</C></Index> <Index Key="rcwa monoids" Subkey="membership test"> <C>rcwa monoids</C> </Index> <Index Key="IsSubset" Subkey="for two rcwa monoids"><C>IsSubset</C></Index> There is a method for <C>Size</C> which computes the order of an rcwa monoid. Further there is a method for <C>in</C> which checks whether a given rcwa mapping lies in a given rcwa monoid (membership test), and there is a method for <C>IsSubset</C> which checks for a submonoid relation. <P/> <Index Key="Support" Subkey="of an rcwa monoid"><C>Support</C></Index> <Index Key="Modulus" Subkey="of an rcwa monoid"><C>Modulus</C></Index> <Index Key="IsTame" Subkey="for an rcwa monoid"><C>IsTame</C></Index> <Index Key="PrimeSet" Subkey="of an rcwa monoid"><C>PrimeSet</C></Index> <Index Key="IsIntegral" Subkey="for an rcwa monoid"><C>IsIntegral</C></Index> <Index Key="IsClassWiseOrderPreserving" Subkey="for an rcwa monoid"> <C>IsClassWiseOrderPreserving</C> </Index> <Index Key="IsSignPreserving" Subkey="for an rcwa monoid"> <C>IsSignPreserving</C> </Index> There are also methods for <C>Support</C>, <C>Modulus</C>, <C>IsTame</C>, <C>PrimeSet</C>, <C>IsIntegral</C>, <C>IsClassWiseOrderPreserving</C> and <C>IsSignPreserving</C> available for rcwa monoids. <P/> <Index Key="rcwa monoid" Subkey="modulus">rcwa monoid</Index> <Index Key="rcwa monoid" Subkey="tame">rcwa monoid</Index> <Index Key="rcwa monoid" Subkey="wild">rcwa monoid</Index> <Index Key="rcwa monoid" Subkey="prime set">rcwa monoid</Index> <Index Key="rcwa monoid" Subkey="integral">rcwa monoid</Index> <Index Key="rcwa monoid" Subkey="class-wise order-preserving"> rcwa monoid </Index> <Index Key="rcwa monoid" Subkey="sign-preserving">rcwa monoid</Index> The <E>support</E> of an rcwa monoid is the union of the supports of its elements. The <E>modulus</E> of an rcwa monoid is the lcm of the moduli of its elements in case such an lcm exists and 0 otherwise. An rcwa monoid is called <E>tame</E> if its modulus is nonzero, and <E>wild</E> otherwise. The <E>prime set</E> of an rcwa monoid is the union of the prime sets of its elements. An rcwa monoid is called <E>integral</E>, <E>class-wise order-preserving</E> or <E>sign-preserving</E> if all of its elements are so. <Example> <![CDATA[ gap> f1 := RcwaMapping([[-1, 1, 1],[ 0,-1, 1]]);; gap> f2 := RcwaMapping([[ 1,-1, 1],[-1,-2, 1],[-1, 2, 1]]);; gap> f3 := RcwaMapping([[ 1, 0, 1],[-1, 0, 1]]);; gap> N := Monoid(f1,f2,f3);; gap> Size(N); 366 gap> List([Monoid(f1,f2),Monoid(f1,f3),Monoid(f2,f3)],Size); [ 96, 6, 66 ] gap> f1*f2*f3 in N; true gap> IsSubset(N,M); false gap> IsSubset(N,Monoid(f1*f2,f3*f2)); true gap> Support(N); Integers gap> Modulus(N); 6 gap> IsTame(N) and IsIntegral(N); true gap> IsClassWiseOrderPreserving(N) or IsSignPreserving(N); false gap> Collected(List(AsList(N),Image)); # The images of the elements of N. [ [ Integers, 2 ], [ 1(2), 2 ], [ Z \ 1(3), 32 ], [ 0(6), 44 ], [ 0(6) U 1(6), 4 ], [ Z \ 4(6) U 5(6), 32 ], [ 0(6) U 2(6), 4 ], [ 0(6) U 5(6), 4 ], [ 1(6), 44 ], [ 1(6) U [ -1 ], 2 ], [ 1(6) U 3(6), 4 ], [ 1(6) U 5(6), 40 ], [ 2(6), 44 ], [ 2(6) U 3(6), 4 ], [ 3(6), 44 ], [ 3(6) U 5(6), 4 ], [ 5(6), 44 ], [ 5(6) U [ 1 ], 2 ], [ [ -5 ], 1 ], [ [ -4 ], 1 ], [ [ -3 ], 1 ], [ [ -1 ], 1 ], [ [ 0 ], 1 ], [ [ 1 ], 1 ], [ [ 2 ], 1 ], [ [ 3 ], 1 ], [ [ 5 ], 1 ], [ [ 6 ], 1 ] ] ]]> </Example> Finite forward orbits under the action of an rcwa monoid can be found by the operation <C>ShortOrbits</C>: <ManSection> <Meth Name="ShortOrbits" Arg="M, S, maxlng" Label="for rcwa monoid, set of points and bound on length"/> <Returns> A list of finite forward orbits of the rcwa monoid <A>M</A> of length at most <A>maxlng</A> which start at points in the set <A>S</A>. </Returns> <Description> <Example> <![CDATA[ gap> ShortOrbits(M,[-5..5],20); [ [ -5, -4, 1, 2, 7, 8 ], [ -3, -2, 1, 2, 5, 6 ], [ -1, 0, 1, 2, 3, 4 ] ] gap> Display(Action(M,last[1])); Monoid( [ Transformation( [ 2, 3, 4, 3, 6, 3 ] ), Transformation( [ 4, 5, 4, 3, 4, 1 ] ) ], ... ) gap> orbs := ShortOrbits(N,[0..10],100); [ [ -5, -4, -3, -1, 0, 1, 2, 3, 5, 6 ], [ -11, -10, -9, -7, -6, -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12 ], [ -17, -16, -15, -13, -12, -11, -10, -9, -7, -6, -5, -4, -3, -1, 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18 ] ] gap> quots := List(orbs,orb->Action(N,orb));; gap> List(quots,Size); [ 268, 332, 366 ] ]]> </Example> </Description> </ManSection> Balls of given radius around an element of an rcwa monoid can be computed by the operation <C>Ball</C>. This operation can also be used for computing forward orbits or subsets of such under the action of an rcwa monoid: <ManSection> <Heading> Ball (for monoid, element and radius or monoid, point, radius and action) </Heading> <Meth Name ="Ball" Arg="M, f, r" Label="for monoid, element and radius"/> <Meth Name ="Ball" Arg="M, p, r, action" Label="for monoid, point, radius and action"/> <Returns> The ball of radius <A>r</A> around the element <A>f</A> in the monoid <A>M</A>, respectively the ball of radius <A>r</A> around the point <A>p</A> under the action <A>action</A> of the monoid <A>M</A>. </Returns> <Description> All balls are understood with respect to <C>GeneratorsOfMonoid(<A>M</A>)</C>. As membership tests can be expensive, the first-mentioned method does not check whether <A>f</A> is indeed an element of <A>M</A>. The methods require that point- / element comparisons are cheap. They are not only applicable to rcwa monoids. If the option <A>Spheres</A> is set, the ball is splitted up and returned as a list of spheres. <Example> <![CDATA[ gap> List([0..12],k->Length(Ball(N,One(N),k))); [ 1, 4, 11, 26, 53, 99, 163, 228, 285, 329, 354, 364, 366 ] gap> Ball(N,[0..3],2,OnTuples); [ [ -3, 3, 3, 3 ], [ -1, -3, 0, 2 ], [ -1, -1, -1, -1 ], [ -1, -1, 1, -1 ], [ -1, 1, 1, 1 ], [ -1, 3, 0, -4 ], [ 0, -1, 2, -3 ], [ 0, 1, 2, 3 ], [ 1, -1, -1, -1 ], [ 1, 3, 0, 2 ], [ 3, -4, -1, 0 ] ] gap> l := 2*IdentityRcwaMappingOfZ; r := l+1; Rcwa mapping of Z: n -> 2n Rcwa mapping of Z: n -> 2n + 1 gap> Ball(Monoid(l,r),1,4,OnPoints:Spheres); [ [ 1 ], [ 2, 3 ], [ 4, 5, 6, 7 ], [ 8, 9, 10, 11, 12, 13, 14, 15 ], [ 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 ] ] ]]> </Example> </Description> </ManSection> <Alt Only="HTML"> </Alt> </Section> <!-- #################################################################### --> </Chapter> <!-- #################################################################### -->