Singular Manual: sort

  | #### D.2.3.14 sort Procedure from library `general.lib` (see general_lib). Usage: sort(id[,v,o,n]); id = ideal/module/intvec/list sort may be called with 1, 2 or 3 arguments in the following way: sort(id[,v,n]); v=intvec of positive integers, n=integer, sort(id[,o,n]); o=string (any allowed ordstr of a ring), n=integer Return: a list l of two elements:
              l[1]: object of same type as input but sorted in the following way:
                 - if id=ideal/module: generators of id are sorted w.r.t. intvec v
                   (id[v[1]] becomes 1-st, id[v[2]]  2-nd element, etc.). If no v is
                   present, id is sorted w.r.t. ordering o (if o is given) or w.r.t.
                   actual monomial ordering (if no o is given):
                   NOTE: generators with SMALLER(!) leading term come FIRST
                   (e.g. sort(id); sorts backwards to actual monomial ordering)
                 - if id=list or intvec: sorted w.r.t. < (indep. of other arguments)
                 - if n!=0 the ordering is inverse, i.e. w.r.t. v(size(v)..1)
                   default: n=0
               l[2]: intvec, describing the permutation of the input (hence l[2]=v
                   if v is given (with positive integers))
Note: If v is given id may be any simply indexed object (e.g. any list or string); if v[i]<0 and i<=size(id) v[i] is set internally to i; entries of v must be pairwise distinct to get a permutation id. Zero generators of ideal/module are deleted If 'o' is given, the input is sorted by considering leading terms w.r.t. the new ring ordering given by 'o' Example:
    LIB "general.lib";
    ring r0 = 0,(x,y,z,t),lp;
    ideal i = x3,z3,xyz;
    sort(i);            //sorts using lex ordering, smaller polys come first
    ==> [1]:
    ==>    _[1]=z3
    ==>    _[2]=xyz
    ==>    _[3]=x3
    ==> [2]:
    ==>    2,3,1
    sort(i,3..1);
    ==> [1]:
    ==>    _[1]=xyz
    ==>    _[2]=z3
    ==>    _[3]=x3
    ==> [2]:
    ==>    3,2,1
    sort(i,"ls")[1];     //sort w.r.t. negative lex ordering
    ==> _[1]=x3
    ==> _[2]=xyz
    ==> _[3]=z3
    intvec v =1,10..5,2..4;v;
    ==> 1,10,9,8,7,6,5,2,3,4
    sort(v)[1];          // sort v lexicographically
    ==> 1,2,3,4,5,6,7,8,9,10
    sort(v,"Dp",1)[1];   // sort v w.r.t (total sum, reverse lex)
    ==> 1,2,3,4,5,6,7,8,9,10
    // Note that in general: lead(sort(M)) != sort(lead(M)), e.g:
    module M = [0, 1, 1, 0], [1, 0, 0, 1]; M;
    ==> M[1]=gen(3)+gen(2)
    ==> M[2]=gen(4)+gen(1)
    sort(lead(M), "c, dp")[1];
    ==> _[1]=gen(4)
    ==> _[2]=gen(3)
    lead(sort(M, "c, dp")[1]);
    ==> _[1]=gen(3)
    ==> _[2]=gen(4)
    // In order to sort M wrt a NEW ordering by considering OLD leading
    // terms use one of the following equivalent commands:
    module( M[ sort(lead(M), "c,dp")[2] ] );
    ==> _[1]=gen(4)+gen(1)
    ==> _[2]=gen(3)+gen(2)
    sort( M, sort(lead(M), "c,dp")[2] )[1];
    ==> _[1]=gen(4)+gen(1)
    ==> _[2]=gen(3)+gen(2)
    // BUG: Please, don't use this sort for integer vectors or lists
    // with them if there can be negative integers!
    // TODO: for some HiWi
    sort(3..-3)[1];
    ==> -3,-2,-1,0,1,2,3
    sort(list(-v, v))[1];
    ==> [1]:
    ==>    -1,-10,-9,-8,-7,-6,-5,-2,-3,-4
    ==> [2]:
    ==>    1,10,9,8,7,6,5,2,3,4

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