Is it a complete set ? How to search matrix?

https://gist.github.com/hoyeunglee/b0c6b61fafd1bee988fcafef3cfe6025
https://gist.github.com/hoyeunglee/a1e0ca335be8bbc8fde301c6aded66f8

test1 = o(a(x,y),z)

test1.args[0]

python sympy can not get function name o and a

restart;
with(Groebner):

DoExist := proc(tau, n)
if rtable_num_elems(tau) >= n then
	return tau[n];
else
	return 0;
end if;
end proc;

IsZero := proc(a, b)
if a=0 then 
	return 0;
else 
	return a/b;
end if
end proc;

g1 := x^2-w*y;
g2 := x*y - w*z;
g3 := y^2 - x*z;
gro := Basis([g1,g2,g3],plex(x,y,z,w));

X1 := `*`(LeadingTerm(g1, tdeg(x, y, z, w)));
X2 := `*`(LeadingTerm(g2, tdeg(x, y, z, w)));
X3 := `*`(LeadingTerm(g3, tdeg(x, y, z, w)));
X12 := lcm(X1,X2);
X13 := lcm(X1,X3);
X23 := lcm(X2,X3);
S12 := SPolynomial(g1, g2, lexdeg([x, y, z, w]));
S23 := SPolynomial(g2, g3, lexdeg([x, y, z, w]));
S13 := SPolynomial(g1, g3, lexdeg([x, y, z, w]));
e1 := Vector([1,0,0]);
e2 := Vector([0,1,0]);
e3 := Vector([0,0,1]);
eq1:= S12 = h121*g1 + h122*g2 + h123*g3;
eq1 := S12 - expand(h121*g1 + h122*g2 + h123*g3):
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h121, h122, h123)), tdeg(x, y, z, w, h121, h122, h123), 'Q');
h121 := 0;
h122 := 0;
h123 := 0;
s12 := IsZero(X12,X1)*e1-IsZero(X12,X2)*e2-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);

eq1 := S13-expand(h131*g1+h132*g2+h133*g3):
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h131, h132, h133)), tdeg(x, y, z, w, h131, h132, h133), 'Q');
h131 := 0;
h132 := 0;
h133 := 0;
s13 := IsZero(X13,X1)*e1-IsZero(X13,X3)*e3-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);

eq1:= S23 - expand(h231*g1 + h232*g2 + h233*g3);
NormalForm(eq1, Basis([g1, g2, g3], tdeg(x, y, z, w, h231, h232, h233)), tdeg(x, y, z, w, h231, h232, h233), 'Q');
h231 := 0;
h232 := 0;
h233 := 0;
s23 := IsZero(X23,X2)*e2-IsZero(X23,X3)*e3-Vector([DoExist(<Q>,3), DoExist(<Q>,2), DoExist(<Q>,1)]);

with(LinearAlgebra):
#F = Syz*GrobnerBasis
F := simplify(MatrixMatrixMultiply(Matrix([[s12[1],s13[1],s23[1]],[s12[2],s13[2],s23[2]],[s12[3],s13[3],s23[3]]]), Matrix([[gro[1]],[gro[2]],[gro[3]]])));
F[1][1] - g1 = 0; 
F[2][1] - g2 = 0;
F[3][1] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[s12[1],0,s23[1]],[s12[2],0,s23[2]],[s12[3],0,s23[3]]]), Matrix([[gro[1]],[gro[2]],[gro[3]]])));
F[1][1] - g1 = 0; 
F[2][1] - g2 = 0;
F[3][1] - g3 = 0;

#F = GrobnerBasis*Syz
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],s13[1],s23[1]],[s12[2],s13[2],s23[2]],[s12[3],s13[3],s23[3]]])));
F[1][1] - g1 = 0; 
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],0,s23[1]],[s12[2],0,s23[2]],[s12[3],0,s23[3]]])));
F[1][1] - g1 = 0; 
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;

#F = GrobnerBasis*Syz
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],s12[1],s12[1]],[s13[2],s13[2],s13[2]],[s23[3],s23[3],s23[3]]])));
F[1][1] - g1 = 0; 
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;
F := simplify(MatrixMatrixMultiply(Matrix([[gro[1],gro[2],gro[3]]]), Matrix([[s12[1],0,s12[1]],[s13[2],0,s13[2]],[s23[3],0,s23[3]]])));
F[1][1] - g1 = 0; 
F[1][2] - g2 = 0;
F[1][3] - g3 = 0;

syz result is s12, s23

but after verify, F is not equal to GrobnerBasis*Syz

How to find the range of poset?

i would like to search the range of range is equal to the range of domain after solve system.

it may not exactly equal but around 80% of elements in range belong to the range of domain

if it is not factorisation domain,

how to generate ideal with function in maple?

is there an algorithm to search whether protocol exist for movement of balls in different boxes

for example

assume there are 5 boxes, each box has a variable number of number balls

ball number 1 move from box 1 to box 3 and box 5

then when discover ball number 8 in any box, ball number 1 will disappear

if input data are 5 list of numbers each row

1. [1,2,3],[5,7,12],[12,14,15],[1,5,7,9,15],[12,13,14]

2. [2,3],[5,7,12],[1,12,14,15],[1,5,7,9,15],[1,12,13,14]

....

etc