#
# file:    o1.gap
# purpose: O_1-related stuff
# created: pasha feb 18 2005
# 


O1 := ReducedPolyRing (GroundField, 1);
O1Basis := Basis (O1); # supposed to be <x^0, x^1, ..., x^{p-1}>

# derivation in O1: f -> df/dx
list := [Zero(O1)];
for i in [1..Dimension(O1)-1] do
	Add (list, i * O1Basis[i]);
od;
D := LeftModuleHomomorphismByImagesNC (O1, O1, O1Basis, list);

# multiplication on x in O1: f -> xf
list := [];
for i in [2..Dimension(O1)] do
	Add (list, O1Basis[i]);
od;
Add (list, Zero(O1));
R := LeftModuleHomomorphismByImagesNC (O1, O1, O1Basis, list);
Unbind (list);


# alternative, less preferable methods to define this
# (they are not linear mappings and we cannnot add them without additional work)

## derivation in O1: f -> df/dx
#d := function (f)
#	local coeff, coeff_result, i;
#	coeff := Coefficients (O1Basis, f);
#	coeff_result := [];
#	for i in [1..Dimension(O1)-1] do
#		coeff_result[i] := i*coeff[i+1];
#	od;
#	coeff_result[Dimension(O1)] := 0;
#	return (LinearCombination (O1Basis, coeff_result));
#end;
#
## multiplication on x in O1: f -> xf
#r := function (f)
#	local coeff, coeff_result, i;
#	coeff := Coefficients (O1Basis, f);
#	coeff_result := [0];
#	for i in [2..Dimension(O1)] do
#		coeff_result[i] := coeff[i-1];
#	od;
#	return (LinearCombination (O1Basis, coeff_result));
#end;
#
#D := MappingByFunction (O1, O1, d);
#R := MappingByFunction (O1, O1, r); 


# end of o1.gap