7.2.2.1 map declarations (plural)

Syntax:

map name = preimage_ring_name , ideal_expression ;
map name = preimage_ring_name , list_of_poly_and_ideal_expressions ;
map name = map_expression ;

Purpose:

defines a ring map from preimage_ring to basering.
Maps the variables of the preimage ring to the generators of the ideal.
If the ideal contains less elements than the number of variables in the preimage_ring, the remaining variables are mapped to 0.
If the ideal contains more elements, extra elements are ignored.
The image ring is always the current basering. For the mapping of coefficients from different fields see map (plural).

Default:

none

Note:

There are standard mappings for maps which are close to the identity map: fetch (plural) and imap (plural).

The name of a map serves as the function which maps objects from the preimage_ring into the basering. These objects must be defined by names (no evaluation in the preimage ring is possible).

Example:

// an easy example ring r1 = 0,(a,b),dp; // a commutative ring poly P = a^2+ab+b^3; ring r2 = 0,(x,y),dp; def W=nc_algebra(1,-1); // a Weyl algebra setring W; map M = r1, x^2, -y^3; // note: M is a map and not a morphism M(P); ==> -y9-x2y3+x4 // now, a more involved example LIB "ncalg.lib"; def Usl2 = makeUsl2(); // this algebra is U(sl_2), generated by e,f,h setring Usl2; poly P = 4*e*f+h^2-2*h; // the central el-t of Usl2 poly Q = e^3*f-h^4; // some polynomial ring W1 = 0,(D,X),dp; def W2=nc_algebra(1,-1); setring W2; // this algebra is the opposite Weyl algebra map F = Usl2, -X, D*D*X, 2*D*X; F(P); // 0, because P is in the kernel of F ==> 0 F(Q); ==> -16D4X4+96D3X3-D2X4-112D2X2+6DX3+16DX-6X2

See fetch (plural); ideal expressions (plural); imap (plural); map (plural); ring (plural).