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D.4.2.8 is_bijective
Procedure from library algebra.lib (see algebra_lib).
- Usage:
- is_bijective(phi,pr); phi map to basering, pr preimage ring
- Return:
- an integer, 1 if phi is bijective, 0 if not
- Note:
- The algorithm checks first injectivity and then surjectivity.
To interprete this for local/mixed orderings, or for quotient rings
type help is_surjective; and help is_injective;
- Display:
- A comment if printlevel >= voice-1 (default)
Example:
| | LIB "algebra.lib";
int p = printlevel; printlevel = 1;
ring R = 0,(x,y,z),dp;
ideal i = x, y, x2-y3;
map phi = R,i; // a map from R to itself, z->x2-y3
is_bijective(phi,R);
==> // map not injective
==> 0
qring Q = std(z-x2+y3);
is_bijective(ideal(x,y,x2-y3),Q);
==> 1
ring S = 0,(a,b,c,d),dp;
map psi = R,ideal(a,a+b,c-a2+b3,0); // a map from R to S,
is_bijective(psi,R); // x->a, y->a+b, z->c-a2+b3
==> // map injective, but not surjective
==> 0
qring T = std(d,c-a2+b3);
==> // ** _ is no standard basis
map chi = Q,a,b,a2-b3; // amap between two quotient rings
is_bijective(chi,Q);
==> 1
printlevel = p;
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