|
#############################################################################
##
#W pseudoFrobenius.gi Manuel Delgado <mdelgado@fc.up.pt>
#W Pedro A. Garcia-Sanchez <pedro@ugr.es>
##
#Y Copyright 2015-- Centro de Matemática da Universidade do Porto, Portugal and Universidad de Granada, Spain
#############################################################################
################################################################
## The main function in this file is
#F NumericalSemigroupsWithPseudoFrobeniusNumbers
## Input: PF (a set of postive integers)
## ouput: a list of numerical semigrups S such that PF(S)=PF
#################################################################
##
## Info classes have been created. It may be used by typing: SetInfoLevel(InfoTipo,2);
##
DeclareInfoClass("InfoTipo");
#################################################################
##
########
## Several auxiliary functions are included
#############################################
#################################################################
#F NumericalSemigroupWithGivenElementsAndFrobenius(elts,frob)
##
## Returns the least numerical semigroup containing the list elts of positive integers and having the largest possible Frobenius number not greater than frob.
## This is just an "abreviation" of NumericalSemigroup(Union(elts,[frob+1..frob+First(elts,IsPosInt)])) that is intended to turn the code more readable.
##
#################################################################
InstallGlobalFunction(NumericalSemigroupWithGivenElementsAndFrobenius, function(elts,frob)
local ns;
if elts <> [] and elts <> [0] then
ns := NumericalSemigroup(Union(elts,[frob+1..frob+Minimum(Difference(elts,[0]))]));
else
ns := NumericalSemigroup([frob+1..2*(frob+1)]);
fi;
return ns;
end);
##
#################################################################
##
#F StartingForcedGapsForPseudoFrobenius(PF)
##
## Computes the set of starting forced gaps, according to the definition in [DG-SR-P15].
##
#################################################################
InstallGlobalFunction(StartingForcedGapsForPseudoFrobenius, function(PF)
local type, forced_gaps, closures, differences, i, x;
type := Length(PF);
forced_gaps := Union([1..Length(PF)],PF); # uses the fact m(S)>=t(S)+1
## justification: lemma:pseudo-comb-pseudo
closures := List([1..type-1], i->SmallElementsOfNumericalSemigroup(NumericalSemigroupWithGivenElementsAndFrobenius(PF{[1..i]},PF[i+1])));
differences := [];
for i in [2..type] do
differences := Union(differences,Filtered(PF[i] - closures[i-1],IsPosInt));
od;
##
forced_gaps := Union(forced_gaps,differences);
##### add divisors
forced_gaps := Union(List(forced_gaps,DivisorsInt));
#####
Info(InfoTipo,2,"starting forced gaps:\n",forced_gaps,"\n");
return forced_gaps;
end);
#################################################################
#################################################################
##
#F FurtherForcedElementsForPseudoFrobenius(f_gaps,f_elts,PF)
##
## computes a set of elements forced by the known forced gaps (f_gaps) and forced elements (f_elts)
## returns the set computed if it is disjoint from f_gaps, returns fail otherwise.
##
## NOTE: "PF" is exactly the set of pseudo-frobenius numbers
##
## the computation is done in two parts (elements forced by exclusion and big elements forced by small gaps)
## The justifications:
## - exclusion: $x\in Z\S$ if and only if $f-x\in S$ for some $f\in PF(S)$
## - big elements: if m is the multiplicity and 1 <= i < m, then frob-i + m \in S. So, either frob-i \in S or frob-i is pseudo-frobenius.
##
#################################################################
InstallGlobalFunction(FurtherForcedElementsForPseudoFrobenius, function(f_gaps,f_elts,PF)
local frob, ef_elts, x, filt, candidates, m, bf_elts, nf_elts, closure,
elts, pf_plus_elements, conflicts;
frob := Maximum(PF);
###exclusion
ef_elts := [];
#( Lemma 10 )
for x in f_gaps do
filt := Filtered(PF, f -> not((f-x in f_gaps) or (f-x < 0)));
if Length(filt) = 1 then
AddSet(ef_elts, filt[1]-x);
fi;
od;
#( Lemma 11 )
candidates := Difference([1..frob-1],Union(f_elts,ef_elts,PF));
for x in candidates do
if Filtered(Difference(PF-x,f_gaps),IsPosInt) = [] then
AddSet(ef_elts, x);
fi;
od;
###big elements (Lemma 12)
m := First(Integers,n-> IsPosInt(n) and not(n in f_gaps)); #least integer that is not a forced gap
bf_elts := Difference(frob - [1..m-1], PF);
### test
nf_elts := Union(f_elts,ef_elts,bf_elts);
closure := NumericalSemigroupWithGivenElementsAndFrobenius(nf_elts,frob);
nf_elts := SmallElementsOfNumericalSemigroup(closure);
nf_elts := Union(nf_elts,[Maximum(nf_elts)..frob+1]);
#############
##added due to an observation of Ignacio Ojeda
elts := Difference(nf_elts,[0,frob+1]);
#From the definition: the sum of a pseudo-Frobenius with an element is an element
pf_plus_elements := Intersection(Union(List(PF,f->f+elts)),[0..frob+1]);
nf_elts := Union(nf_elts,pf_plus_elements);
##############
conflicts := Intersection(f_gaps,nf_elts);
if conflicts = [] then
return nf_elts;
else
Info(InfoTipo,2,"There is no numerical semigroup with the given set as set of pseudo-Frobenius numbers, since the integers ",conflicts," would have to be at the same time forced gaps and forced integers.\n");
return fail;
fi;
end);
#################################################################
#################################################################
##
#F FurtherForcedGapsForPseudoFrobenius(f_gaps,f_elts,PF)
##
## Returns a list of integers that must be gaps of any numerical semigroup containing elts and for which it is known that f_gaps are gaps
## Returns fail in case it finds an element that had to be a gap, which implies that no semigroup exists having the given sets of gaps and elements.
##
## Justification: note that if f is a gap and e is an element, then f-e is a gap. (Otherwise f=(f-e)+e would be an element)
##
#################################################################
InstallGlobalFunction(FurtherForcedGapsForPseudoFrobenius, function(f_gaps,f_elts,PF)
local frob, elts, nf_gaps, conflicts;
frob := Maximum(PF);
elts := Difference(f_elts,[0,frob+1]);
nf_gaps := Union(List(f_gaps, g -> g - elts));
nf_gaps := Filtered(nf_gaps,IsPosInt);
nf_gaps := Union(List(nf_gaps,DivisorsInt));
##test
conflicts := Intersection(nf_gaps,f_elts);
if conflicts = [] then
return Union(f_gaps,nf_gaps);
else
Info(InfoTipo,2,"There is no numerical semigroup with the given set as set of pseudo-Frobenius numbers, since the integers",conflicts," would have to be at the same time forced gaps and forced integers.\n");
return fail;
fi;
end);
#################################################################
##
#F SimpleForcedIntegersForPseudoFrobenius(f_gaps,f_elts,PF)
##
## The aim of this function is to compute forced gaps and forced integers for a semigroup having PF as set of pseudo-Frobenius numbers, containing f_elts and having f_gaps as some of its gaps.
##
## The function consists of a loop that makes use of the functions "FurtherForcedGapsForPseudoFrobenius" and "FurtherForcedElementsForPseudoFrobenius" to discover new forced gaps and new forced elements, respectively.
##
## If it is discovered an integers that simoultaniously had to be a gap and an element, we say that there is a conflict and "fail" is returned. Otherwise, the function returns a pair [forced_gaps, forced_elements].
#################################################################
InstallGlobalFunction(SimpleForcedIntegersForPseudoFrobenius, function(f_gaps,f_elts,PF)
local nf_gaps, nf_elts, changes, gaps, elts;
nf_gaps := ShallowCopy(f_gaps);
nf_elts := ShallowCopy(f_elts);
repeat
changes := false;
gaps := FurtherForcedGapsForPseudoFrobenius(nf_gaps,nf_elts,PF);
if gaps = fail then
return fail;
elif gaps <> nf_gaps then
changes := true;
nf_gaps := gaps;
fi;
elts := FurtherForcedElementsForPseudoFrobenius(nf_gaps,nf_elts,PF);
if elts = fail then
return fail;
elif elts <> nf_elts then
changes := true;
nf_elts := elts;
fi;
until not changes;
return [nf_gaps,nf_elts];
end);
#################################################################
##
#F NonAdmissibleForPseudoFrobenius(f_gaps,f_elts,PF)
##
## determination of new forced gaps based on the concept of admissible integer
##
## input: a pair of sets of forced gaps and forced elements
## output: non-admissible integers, which (by Lemma~\ref{lemma:admissible_ints}) are new forced gaps.
#################################################################
InstallGlobalFunction(NonAdmissibleForPseudoFrobenius, function(f_gaps,f_elts,PF)
local frob, admissible, totest, v, pnf_ce, non_admissible;
frob := Maximum(PF);
admissible := [];
totest := Difference([1..frob], Union(f_gaps,f_elts));
while totest <> [] do
v := totest[1];
pnf_ce:= SimpleForcedIntegersForPseudoFrobenius(f_gaps,Union(f_elts,[v]),PF);
if pnf_ce <> fail then
admissible := Union(admissible,pnf_ce[2]);
totest := Difference(totest,admissible);
else
totest := Difference(totest,[v]);
fi;
od;
non_admissible := Difference([1..frob], admissible);
Info(InfoTipo,2,"Non admissible integers\n",non_admissible,"\n");
return non_admissible;
end);
#################################################################
##
## Let PF={g_1<...<g_n} be a set of integers such that there exists a numerical semigroup S such that PF(S)=PF.
## The aim of this function is to compute forced integers (gaps or elements) for any semigroup S such that PF(S)=PF.
##
##
#F ForcedIntegersForPseudoFrobenius(PF)
###
# Input:
# * one list of integers: [g_1,...,g_n]
# or
# * integers: g_1,...,g_n
#
###
# Output:
## * in case there exists a numerical semigroup S such that PF(S)=PF:
## - a list [forced_gaps,forced_elts] such that:
## -- forced_gaps is contained in N\S for any numerical semigroup S such that PF(S)={g_1,...,g_n}
## -- forced_elts is contained in S for any numerical semigroup S such that PF(S)={g_1,...,g_n}
## * "fail" in case it is found some condition that fails
#################################################################
InstallGlobalFunction(ForcedIntegersForPseudoFrobenius, function(arg)
local PF, type, frob, sfg, f_ints, n_ad, new_gaps;
## arguments
if Length(arg) = 1 and IsList(arg[1]) then
PF := Set(arg[1]);
else
PF := Set(arg);
fi;
##
type := Length(PF);
frob := Maximum(PF);
##
if type = 1 then
if IsEvenInt(frob) then # frob/2 is also a pseudo-Frobenius number, thus the type can not be 1.
return [];
fi;
return [DivisorsInt(frob),[0,frob+1]];
fi;
sfg := StartingForcedGapsForPseudoFrobenius(PF);
f_ints := SimpleForcedIntegersForPseudoFrobenius(sfg,[],PF);
if f_ints = fail then
return fail;
elif IsRange(Union(f_ints)) then
#a test (motivated by a problem found by Ojeda)
if PseudoFrobeniusOfNumericalSemigroup(NumericalSemigroupByGaps(f_ints[1])) <> PF then
Error("There is a problem in ForcedIntegersForPseudoFrobenius. Please communicate the input to the numericalsgps package authors");
fi;
return f_ints;
fi;
n_ad := NonAdmissibleForPseudoFrobenius(f_ints[1],f_ints[2],PF);
new_gaps := Difference(n_ad,f_ints[1]);
Info(InfoTipo,2,"extra forced gaps\n",new_gaps,"\n");
return SimpleForcedIntegersForPseudoFrobenius(Union(new_gaps,f_ints[1]),f_ints[2],PF);
end);
#############################################
## a quick version
InstallGlobalFunction(ForcedIntegersForPseudoFrobenius_QV, function(arg)
local PF, type, frob, sfg;
## arguments
if Length(arg) = 1 and IsList(arg[1]) then
PF := Set(arg[1]);
else
PF := Set(arg);
fi;
##
type := Length(PF);
frob := Maximum(PF);
##
if type = 1 then
return [DivisorsInt(frob),[0,frob+1]];
fi;
##
sfg := StartingForcedGapsForPseudoFrobenius(PF);
return SimpleForcedIntegersForPseudoFrobenius(sfg,[],PF);
end);
#################################################################
#################################################################
#################################################################
##
#F NumericalSemigroupsWithPseudoFrobeniusNumbers(PF)
## Input: PF (a set of postive integers)
## Ouput: a list of numerical semigrups S such that PF(S)=PF
## When Length(PF)=1, it makes use of the function NumericalSemigroupsWithFrobeniusNumber
##
## The option draw_tree may be useful, since it gives an output that gives some light on the way the computations are done
##
## example: NumericalSemigroupsWithPseudoFrobeniusNumbers(rec(pseudo_frobenius := pf, draw_tree := true));
#################################################################
InstallGlobalFunction(NumericalSemigroupsWithPseudoFrobeniusNumbers, function(arg)
local PF, type, frob, initially_forced_integers,
freeElementsTree_recursive, semigroups, fg, fe;
## arguments
if Length(arg) = 1 and IsList(arg[1]) then
PF := Set(arg[1]);
else
PF := Set(arg);
fi;
##
type := Length(PF);
frob := Maximum(PF);
if type = 1 then
if IsEvenInt(PF[1]) then # PF[1]/2 is also a pseudo-Frobenius number, thus the type can not be 1.
return [];
fi;
Info(InfoTipo,2, "As the type is 1, the function IrreducibleNumericalSemigroupsWithFrobeniusNumber will be used");
return IrreducibleNumericalSemigroupsWithFrobeniusNumber(frob);
elif type = 2 and 2*PF[1] = PF[2] then
Info(InfoTipo,2, "As the type is 2 and frob is even, the function IrreducibleNumericalSemigroupsWithFrobeniusNumber will be used");
return IrreducibleNumericalSemigroupsWithFrobeniusNumber(frob);
fi;
## testing some simple conditions
if PF[type] - PF[type-1] > PF[1] then
return [];
fi;
initially_forced_integers := ForcedIntegersForPseudoFrobenius(PF);
if initially_forced_integers = fail then
return [];
fi;
##################### local recursive function #####################
# rightmost indicates if the visiting node is the rightmost of the current free integers
freeElementsTree_recursive := function(fg,fe)
local forced_integers, free, ending_condition, nfg, current_free, v,
left, right;
forced_integers := Union(fg,fe);
free := Difference([1..frob], forced_integers);
########################## local function ##########################
## Ending condition ##
## The number of free elements must be <= 1
ending_condition := function(g,e)
local forced_integers, free;
# Error("..");
forced_integers := Union(g,e);
free := Difference([1..frob], forced_integers);
if Length(free) = 0 then
#g are the gaps
#e are the small elements
if First(Difference(g,PF), pf -> Intersection(pf + Difference(e,[0]),g) = []) = fail then
Add(semigroups, NumericalSemigroupByGaps(g));
fi;
return;
fi;
if Length(free) = 1 then
if RepresentsGapsOfNumericalSemigroup(g) then
#g are the gaps
#eUfree are the small elements
if First(Difference(g,PF), pf -> Intersection(pf + Difference(Union(e,free),[0]),g) = []) = fail then
Add(semigroups, NumericalSemigroupByGaps(g));
fi;
fi;
if RepresentsGapsOfNumericalSemigroup(Union(g,free)) then
#gUfree are the gaps
#e are the small elements
if First(Difference(Union(g,free),PF), pf -> Intersection(pf + Difference(e,[0]),Union(g,free)) = []) = fail then
Add(semigroups, NumericalSemigroupByGaps(Union(g,free)));
fi;
fi;
return;
fi;
end;
### end of local function ######
##
if Length(free) <= 1 then
ending_condition(fg,fe);
fi;
nfg := ShallowCopy(fg); #used to store gaps...
current_free := ShallowCopy(free);
## the cycle...
while Length(current_free) > 1 do
v := current_free[1];
left := SimpleForcedIntegersForPseudoFrobenius(nfg,Union(fe,[v]),PF);
if left = fail then
right := SimpleForcedIntegersForPseudoFrobenius(Union(nfg,[v]),fe,PF);
if (right = fail) or (Intersection(right[1],right[2]) <> []) then
break;
fi;
else
freeElementsTree_recursive(left[1],left[2]);
fi;
nfg := Union(nfg,[v]);
current_free := Difference(current_free,[v]);
od;
if Length(current_free) = 1 then
ending_condition(nfg,fe);
fi;
end;
########### end of local recursive function #####
semigroups := [];
fg := initially_forced_integers[1];
fe := initially_forced_integers[2];
freeElementsTree_recursive(fg,fe);
return semigroups;
end);
#################################################################
##
#F ANumericalSemigroupWithPseudoFrobeniusNumbers(arg)
## Input: PF (a set of postive integers)
# Alternativelly
# * a record with fields "pseudo_frobenius" and "max_attempts" option
##
## ouput: A numerical semigrups S such that PF(S)=PF. Returns fail if it conludes that it does not exist and suggests to use NumericalSemigroupsWithPseudoFrobeniusNumbers if it is not able to conclude...
## When Length(PF)=1 or Length(PF)=2 and 2*PF[1] = PF[2], it makes use of the function AnIrreducibleNumericalSemigroupWithFrobeniusNumber
#################################################################
InstallGlobalFunction(ANumericalSemigroupWithPseudoFrobeniusNumbers, function(arg)
local m_att, PF, type, frob, of_ints, free, nspfn, i, f_ints, v, nfig,
nfie;
## check arguments
m_att := fail;
if Length(arg) = 1 then
if IsRecord(arg[1]) then
if IsBound(arg[1].pseudo_frobenius) then
PF := Set(arg[1].pseudo_frobenius);
fi;
if IsBound(arg[1].max_attempts) then
m_att := arg[1].max_attempts;
fi;
elif IsInt(arg[1]) then
PF := arg;
else
PF := Set(arg[1]);
fi;
elif Length(arg) > 1 then
PF := Set(arg);
else
Error("please check the arguments...");
fi;
type := Length(PF);
frob := Maximum(PF);
if (type = 1) or (type = 2 and 2*PF[1] = PF[2]) then
Info(InfoTipo,1, "As the type is 1 (or 2 and frob is even), the function AnIrreducibleNumericalSemigroupWithFrobeniusNumber will be used");
return AnIrreducibleNumericalSemigroupWithFrobeniusNumber(frob);
fi;
## testing some simple conditions
if PF[type] - PF[type-1] > PF[1] then
return fail;
fi;
#maximum number of attempts
if m_att = fail then
m_att := Minimum(7,Int(frob/3)); #many experiments suggest that it is a reasonable number
fi;
# forced integers
of_ints := ForcedIntegersForPseudoFrobenius(PF);
if of_ints = fail then
return fail;
fi;
free := Difference([1..frob],Union(of_ints));
if Length(free) < Minimum(10,Int(frob/5)) then #NumericalSemigroupsWithPseudoFrobeniusNumbers is reasonably fast...
nspfn := NumericalSemigroupsWithPseudoFrobeniusNumbers(PF);
if nspfn = [] then
return fail;
else
return RandomList(nspfn);
fi;
fi;
for i in [1..m_att] do
f_ints := ShallowCopy(of_ints);
free := Difference([1..frob],Union(f_ints));
while free <> [] do
v := RandomList(free);
nfig := SimpleForcedIntegersForPseudoFrobenius(Union(f_ints[1],[v]),f_ints[2],PF);
nfie := SimpleForcedIntegersForPseudoFrobenius(f_ints[1],Union(f_ints[2],[v]),PF);
if nfig <> fail then
if IsRange(Union(nfig)) then
return NumericalSemigroupByGaps(nfig[1]);
fi;
f_ints := nfig;
free := Difference([1..frob],Union(f_ints));
elif nfie <> fail then
if IsRange(Union(nfie)) then
return NumericalSemigroupByGaps(nfie[1]);
fi;
f_ints := nfie;
free := Difference([1..frob],Union(f_ints));
else
break;
fi;
Info(InfoTipo,1,"Length free: ",Length(free),"\n");
od;
Info(InfoTipo,1,"Attempt ",i,"\n");
od;
Info(InfoWarning,1,"I have not been able to determine a semigroup satisfying the required conditions. Please increase the number of maximum attempts or use the function NumericalSemigroupsWithPseudoFrobeniusNumbers instead...");
return fail;
end);
[ Seitenstruktur0.42Drucken
etwas mehr zur Ethik
]
|
