[1X4. Residue-Class-Wise Affine Monoids[0X In this short chapter, we describe how to compute with residue-class-wise affine monoids. [13XResidue-class-wise affine[0X monoids, or [13Xrcwa[0X monoids for short, are monoids whose elements are residue-class-wise affine mappings. [1X4.1 Constructing residue-class-wise affine monoids[0X As any other monoids in [5XGAP[0X, residue-class-wise affine monoids can be constructed by [10XMonoid[0X or [10XMonoidByGenerators[0X. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> M := Monoid(RcwaMapping([[ 0,1,1],[1,1,1]]),[0X [4X> RcwaMapping([[-1,3,1],[0,2,1]]));[0X [4X<rcwa monoid over Z with 2 generators>[0X [4Xgap> Size(M);[0X [4X11[0X [4Xgap> Display(MultiplicationTable(M));[0X [4X[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 ], [0X [4X [ 2, 8, 5, 11, 8, 3, 10, 5, 2, 8, 5 ], [0X [4X [ 3, 10, 11, 5, 5, 5, 8, 8, 8, 2, 3 ], [0X [4X [ 4, 9, 6, 8, 8, 8, 5, 5, 5, 7, 4 ], [0X [4X [ 5, 8, 5, 8, 8, 8, 5, 5, 5, 8, 5 ], [0X [4X [ 6, 7, 4, 8, 8, 8, 5, 5, 5, 9, 6 ], [0X [4X [ 7, 5, 8, 6, 5, 4, 9, 8, 7, 5, 8 ], [0X [4X [ 8, 5, 8, 5, 5, 5, 8, 8, 8, 5, 8 ], [0X [4X [ 9, 5, 8, 4, 5, 6, 7, 8, 9, 5, 8 ], [0X [4X [ 10, 8, 5, 3, 8, 11, 2, 5, 10, 8, 5 ], [0X [4X [ 11, 2, 3, 5, 5, 5, 8, 8, 8, 10, 11 ] ][0X [4X[0X [4X------------------------------------------------------------------[0X There are methods for the operations [10XView[0X, [10XDisplay[0X, [10XPrint[0X and [10XString[0X which are applicable to rcwa monoids. All rcwa monoids over a ring R are submonoids of Rcwa(R). The monoid Rcwa(R) itself is not finitely generated, thus cannot be constructed as described above. It is handled as a special case: [1X4.1-1 Rcwa[0X [2X> Rcwa( [0X[3XR[0X[2X ) ________________________________________________________[0Xfunction [6XReturns:[0X The monoid Rcwa([3XR[0X) of all residue-class-wise affine mappings of the ring [3XR[0X. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> RcwaZ := Rcwa(Integers);[0X [4XRcwa(Z)[0X [4Xgap> IsSubset(RcwaZ,M);[0X [4Xtrue[0X [4X[0X [4X------------------------------------------------------------------[0X In our methods to construct rcwa groups, two kinds of mappings played a crucial role, namely the restriction monomorphisms (cf. [2XRestriction[0X ([14X3.1-6[0X)) and the induction epimorphisms (cf. [2XInduction[0X ([14X3.1-7[0X)). The restriction monomorphisms extend in a natural way to the monoids Rcwa(R), and the induction epimorphisms have corresponding generalizations, also. Therefore the operations [10XRestriction[0X and [10XInduction[0X can be applied to rcwa monoids as well: [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> M2 := Restriction(M,2*One(Rcwa(Integers)));[0X [4X<rcwa monoid over Z with 2 generators, of size 11>[0X [4Xgap> Support(M2);[0X [4X0(2)[0X [4Xgap> Action(M2,ResidueClass(1,2));[0X [4XTrivial rcwa group over Z[0X [4Xgap> Induction(M2,2*One(Rcwa(Integers))) = M;[0X [4Xtrue[0X [4X[0X [4X------------------------------------------------------------------[0X [1X4.2 Computing with residue-class-wise affine monoids[0X There is a method for [10XSize[0X which computes the order of an rcwa monoid. Further there is a method for [10Xin[0X which checks whether a given rcwa mapping lies in a given rcwa monoid (membership test), and there is a method for [10XIsSubset[0X which checks for a submonoid relation. There are also methods for [10XSupport[0X, [10XModulus[0X, [10XIsTame[0X, [10XPrimeSet[0X, [10XIsIntegral[0X, [10XIsClassWiseOrderPreserving[0X and [10XIsSignPreserving[0X available for rcwa monoids. The [13Xsupport[0X of an rcwa monoid is the union of the supports of its elements. The [13Xmodulus[0X 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 [13Xtame[0X if its modulus is nonzero, and [13Xwild[0X otherwise. The [13Xprime set[0X of an rcwa monoid is the union of the prime sets of its elements. An rcwa monoid is called [13Xintegral[0X, [13Xclass-wise order-preserving[0X or [13Xsign-preserving[0X if all of its elements are so. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> f1 := RcwaMapping([[-1, 1, 1],[ 0,-1, 1]]);;[0X [4Xgap> f2 := RcwaMapping([[ 1,-1, 1],[-1,-2, 1],[-1, 2, 1]]);; [0X [4Xgap> f3 := RcwaMapping([[ 1, 0, 1],[-1, 0, 1]]);; [0X [4Xgap> N := Monoid(f1,f2,f3);;[0X [4Xgap> Size(N);[0X [4X366[0X [4Xgap> List([Monoid(f1,f2),Monoid(f1,f3),Monoid(f2,f3)],Size);[0X [4X[ 96, 6, 66 ][0X [4Xgap> f1*f2*f3 in N;[0X [4Xtrue[0X [4Xgap> IsSubset(N,M);[0X [4Xfalse[0X [4Xgap> IsSubset(N,Monoid(f1*f2,f3*f2));[0X [4Xtrue[0X [4Xgap> Support(N);[0X [4XIntegers[0X [4Xgap> Modulus(N);[0X [4X6[0X [4Xgap> IsTame(N) and IsIntegral(N);[0X [4Xtrue[0X [4Xgap> IsClassWiseOrderPreserving(N) or IsSignPreserving(N);[0X [4Xfalse[0X [4Xgap> Collected(List(AsList(N),Image)); # The images of the elements of N.[0X [4X[ [ Integers, 2 ], [ 1(2), 2 ], [ Z \ 1(3), 32 ], [ 0(6), 44 ], [0X [4X [ 0(6) U 1(6), 4 ], [ Z \ 4(6) U 5(6), 32 ], [ 0(6) U 2(6), 4 ], [0X [4X [ 0(6) U 5(6), 4 ], [ 1(6), 44 ], [ 1(6) U [ -1 ], 2 ], [0X [4X [ 1(6) U 3(6), 4 ], [ 1(6) U 5(6), 40 ], [ 2(6), 44 ], [0X [4X [ 2(6) U 3(6), 4 ], [ 3(6), 44 ], [ 3(6) U 5(6), 4 ], [ 5(6), 44 ], [0X [4X [ 5(6) U [ 1 ], 2 ], [ [ -5 ], 1 ], [ [ -4 ], 1 ], [ [ -3 ], 1 ], [0X [4X [ [ -1 ], 1 ], [ [ 0 ], 1 ], [ [ 1 ], 1 ], [ [ 2 ], 1 ], [ [ 3 ], 1 ], [0X [4X [ [ 5 ], 1 ], [ [ 6 ], 1 ] ][0X [4X[0X [4X------------------------------------------------------------------[0X Finite forward orbits under the action of an rcwa monoid can be found by the operation [10XShortOrbits[0X: [1X4.2-1 ShortOrbits[0X [2X> ShortOrbits( [0X[3XM, S, maxlng[0X[2X ) ________________________________________[0Xmethod [6XReturns:[0X A list of finite forward orbits of the rcwa monoid [3XM[0X of length at most [3Xmaxlng[0X which start at points in the set [3XS[0X. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> ShortOrbits(M,[-5..5],20);[0X [4X[ [ -5, -4, 1, 2, 7, 8 ], [ -3, -2, 1, 2, 5, 6 ], [ -1, 0, 1, 2, 3, 4 ] ][0X [4Xgap> Display(Action(M,last[1]));[0X [4XMonoid( [ Transformation( [ 2, 3, 4, 3, 6, 3 ] ), [0X [4X Transformation( [ 4, 5, 4, 3, 4, 1 ] ) ], ... )[0X [4Xgap> orbs := ShortOrbits(N,[0..10],100);[0X [4X[ [ -5, -4, -3, -1, 0, 1, 2, 3, 5, 6 ], [0X [4X [ -11, -10, -9, -7, -6, -5, -4, -3, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, [0X [4X 11, 12 ], [0X [4X [ -17, -16, -15, -13, -12, -11, -10, -9, -7, -6, -5, -4, -3, -1, 0, 1, [0X [4X 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18 ] ][0X [4Xgap> quots := List(orbs,orb->Action(N,orb));;[0X [4Xgap> List(quots,Size);[0X [4X[ 268, 332, 366 ][0X [4X[0X [4X------------------------------------------------------------------[0X Balls of given radius around an element of an rcwa monoid can be computed by the operation [10XBall[0X. This operation can also be used for computing forward orbits or subsets of such under the action of an rcwa monoid: [1X4.2-2 Ball (for monoid, element and radius or monoid, point, radius and[0X [1Xaction)[0X [2X> Ball( [0X[3XM, f, r[0X[2X ) ____________________________________________________[0Xmethod [2X> Ball( [0X[3XM, p, r, action[0X[2X ) ____________________________________________[0Xmethod [6XReturns:[0X The ball of radius [3Xr[0X around the element [3Xf[0X in the monoid [3XM[0X, respectively the ball of radius [3Xr[0X around the point [3Xp[0X under the action [3Xaction[0X of the monoid [3XM[0X. All balls are understood with respect to [10XGeneratorsOfMonoid([3XM[0X)[0X. As membership tests can be expensive, the first-mentioned method does not check whether [3Xf[0X is indeed an element of [3XM[0X. The methods require that point- / element comparisons are cheap. They are not only applicable to rcwa monoids. If the option [3XSpheres[0X is set, the ball is splitted up and returned as a list of spheres. [4X--------------------------- Example ----------------------------[0X [4X[0X [4Xgap> List([0..12],k->Length(Ball(N,One(N),k)));[0X [4X[ 1, 4, 11, 26, 53, 99, 163, 228, 285, 329, 354, 364, 366 ][0X [4Xgap> Ball(N,[0..3],2,OnTuples);[0X [4X[ [ -3, 3, 3, 3 ], [ -1, -3, 0, 2 ], [ -1, -1, -1, -1 ], [0X [4X [ -1, -1, 1, -1 ], [ -1, 1, 1, 1 ], [ -1, 3, 0, -4 ], [ 0, -1, 2, -3 ], [0X [4X [ 0, 1, 2, 3 ], [ 1, -1, -1, -1 ], [ 1, 3, 0, 2 ], [ 3, -4, -1, 0 ] ][0X [4Xgap> l := 2*IdentityRcwaMappingOfZ; r := l+1;[0X [4XRcwa mapping of Z: n -> 2n[0X [4XRcwa mapping of Z: n -> 2n + 1[0X [4Xgap> Ball(Monoid(l,r),1,4,OnPoints:Spheres);[0X [4X[ [ 1 ], [ 2, 3 ], [ 4, 5, 6, 7 ], [ 8, 9, 10, 11, 12, 13, 14, 15 ], [0X [4X [ 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 ] ][0X [4X[0X [4X------------------------------------------------------------------[0X