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D.4.1.8 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 interpret 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 standardbasis map chi = Q,a,b,a2-b3; // amap between two quotient rings is_bijective(chi,Q); → 1 printlevel = p; |