{
 "cells": [{
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Numerical semigroups with `francy`\n",
    "\n",
    "## Load `francy` Package"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [{
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "#I  Please load package NormalizInterface or 4ti2Interface\n",
      "#I  to have extended functionalities.\n",
      "#I  Please load package SingularInterface or singular (not both)\n",
      "#I  or GradedModules to have extended functionalities.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 1,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    },
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 2,
     "metadata": {
      "text/plain": ""
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LoadPackage(\"francy\");\n",
    "LoadPackage(\"num\");"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drawing Apéry sets"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example draws the Apéry set of a numerical semigrup with respect to its multiplicity. By passing over a node with the mouse, the set of factorizations with respect to the minimal generating system of the numerical semigroup is displayed. Clicking a node produces a message with the same information."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( arg... ) ... end"
     ]
    },
    "execution_count": 3,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "apery:=function(arg)\n",
    "    local ap,c,hasse, s, n, r, graphHasse, aps, es, canvas, i, order, showfacts, message;\n",
    "    # rel is a list of lists with two elements representin a binary relation\n",
    "    # hasse(rel) removes from rel the pairs [x,y] such that there exists\n",
    "    # z with [x,z],[z,y] in rel\n",
    "    hasse:=function(rel)\n",
    "      local dom, out;\n",
    "      dom:=Flat(rel);\n",
    "      out:=Filtered(rel, p-> ForAny(dom, x->([p[1],x] in rel) and ([x,p[2]] in rel)));\n",
    "      return Difference(rel,out);\n",
    "    end;\n",
    "\n",
    "    order:=function(x)\n",
    "        return Maximum(LengthsOfFactorizationsElementWRTNumericalSemigroup(x,s));\n",
    "    end;\n",
    "\n",
    "    showfacts:=function(x)\n",
    "        message := FrancyMessage(Concatenation(String(x), \" factors as \"), \n",
    "                    String(FactorizationsElementWRTNumericalSemigroup(x,s)));\n",
    "        SetFrancyId(message, Concatenation(\"message-for-\", String(x)));\n",
    "        Add(canvas, message);\n",
    "        return Draw(canvas);\n",
    "    end;\n",
    "    if Length(arg)=1 then\n",
    "        s:=arg[1];\n",
    "        n:=MultiplicityOfNumericalSemigroup(s);\n",
    "    fi;\n",
    "    if Length(arg)=2 then\n",
    "        s:=arg[1];\n",
    "        n:=arg[2];\n",
    "    fi;\n",
    "    if Length(arg)>2 then\n",
    "        Error(\"The number of arguments must be one or two\");\n",
    "    fi;\n",
    "  \n",
    "    graphHasse := Graph(GraphType.UNDIRECTED);\n",
    "    #SetSimulation(graphHasse,true);\n",
    "    #SetDrag(graphHasse,true);\n",
    "    ap:=AperyList(s,n);\n",
    "    c:=Cartesian([1..n],[1..n]);\n",
    "    c:=Filtered(c, p-> ap[p[2]]<>ap[p[1]]);\n",
    "    c:=Filtered(c, p-> ap[p[1]]-ap[p[2]] in s);\n",
    "    c:=hasse(c);\n",
    "    aps:=[];\n",
    "    for i in [1..n] do\n",
    "        aps[i]:=Shape(ShapeType!.CIRCLE, String(ap[i]));\n",
    "        SetLayer(aps[i],-order(ap[i]));\n",
    "        Add(aps[i],Callback(showfacts,[ap[i]]));\n",
    "        Add(aps[i],FrancyMessage(Concatenation(\"{\",\n",
    "            JoinStringsWithSeparator(List(FactorizationsElementWRTNumericalSemigroup(ap[i],s), \n",
    "            f->Concatenation(\"(\",JoinStringsWithSeparator(f,\",\"),\")\")),\",\"),\")\")));\n",
    "        Add(graphHasse,aps[i]);\n",
    "    od;\n",
    "    for r in c do\n",
    "        Add(graphHasse,Link(aps[r[1]],aps[r[2]]));\n",
    "    od;\n",
    "    canvas:=Canvas(\"Apery\");\n",
    "    Add(canvas,graphHasse);\n",
    "    return Draw(canvas);    \n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "scrolled": false
   },
   "outputs": [{
    "data": {
     "application/vnd.francy+json": "{\"version\" : \"1.0.2\",\"mime\" : \"application\\/vnd.francy+json\",\"canvas\" : {\"width\" : 800,\"id\" : \"F45\",\"height\" : 600,\"title\" : \"Apery\",\"zoomToFit\" : true,\"texTypesetting\" : false,\"menus\" : {},\"graph\" : {\"type\" : \"undirected\",\"id\" : \"F1\",\"simulation\" : true,\"collapsed\" : true,\"nodes\" : {\"F2\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F2\",\"size\" : 10,\"title\" : \"0\",\"color\" : \"\",\"layer\" : 0,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F4\" : {\"type\" : \"default\",\"id\" : \"F4\",\"text\" : \"{(0,0,0))\",\"title\" : \"\"}},\"callbacks\" : {\"F3\" : {\"func\" : \"unknown\",\"id\" : \"F3\",\"trigger\" : \"click\",\"knownArgs\" : [\"0\"],\"requiredArgs\" : {}}}},\"F5\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F5\",\"size\" : 10,\"title\" : \"31\",\"color\" : \"\",\"layer\" : -1,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F7\" : {\"type\" : \"default\",\"id\" : \"F7\",\"text\" : \"{(0,0,1))\",\"title\" : \"\"}},\"callbacks\" : {\"F6\" : {\"func\" : \"unknown\",\"id\" : \"F6\",\"trigger\" : \"click\",\"knownArgs\" : [\"31\"],\"requiredArgs\" : {}}}},\"F8\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F8\",\"size\" : 10,\"title\" : \"62\",\"color\" : \"\",\"layer\" : -2,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F10\" : {\"type\" : \"default\",\"id\" : \"F10\",\"text\" : \"{(0,0,2))\",\"title\" : \"\"}},\"callbacks\" : {\"F9\" : {\"func\" : \"unknown\",\"id\" : \"F9\",\"trigger\" : \"click\",\"knownArgs\" : [\"62\"],\"requiredArgs\" : {}}}},\"F11\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F11\",\"size\" : 10,\"title\" : \"93\",\"color\" : \"\",\"layer\" : -3,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F13\" : {\"type\" : \"default\",\"id\" : \"F13\",\"text\" : \"{(0,0,3))\",\"title\" : \"\"}},\"callbacks\" : {\"F12\" : {\"func\" : \"unknown\",\"id\" : \"F12\",\"trigger\" : \"click\",\"knownArgs\" : [\"93\"],\"requiredArgs\" : {}}}},\"F14\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F14\",\"size\" : 10,\"title\" : \"54\",\"color\" : \"\",\"layer\" : -2,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F16\" : {\"type\" : \"default\",\"id\" : \"F16\",\"text\" : \"{(0,2,0))\",\"title\" : \"\"}},\"callbacks\" : {\"F15\" : {\"func\" : \"unknown\",\"id\" : \"F15\",\"trigger\" : \"click\",\"knownArgs\" : [\"54\"],\"requiredArgs\" : {}}}},\"F17\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F17\",\"size\" : 10,\"title\" : \"85\",\"color\" : \"\",\"layer\" : -3,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F19\" : {\"type\" : \"default\",\"id\" : \"F19\",\"text\" : \"{(0,2,1))\",\"title\" : \"\"}},\"callbacks\" : {\"F18\" : {\"func\" : \"unknown\",\"id\" : \"F18\",\"trigger\" : \"click\",\"knownArgs\" : [\"85\"],\"requiredArgs\" : {}}}},\"F20\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F20\",\"size\" : 10,\"title\" : \"116\",\"color\" : \"\",\"layer\" : -4,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F22\" : {\"type\" : \"default\",\"id\" : \"F22\",\"text\" : \"{(0,2,2))\",\"title\" : \"\"}},\"callbacks\" : {\"F21\" : {\"func\" : \"unknown\",\"id\" : \"F21\",\"trigger\" : \"click\",\"knownArgs\" : [\"116\"],\"requiredArgs\" : {}}}},\"F23\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F23\",\"size\" : 10,\"title\" : \"27\",\"color\" : \"\",\"layer\" : -1,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F25\" : {\"type\" : \"default\",\"id\" : \"F25\",\"text\" : \"{(0,1,0))\",\"title\" : \"\"}},\"callbacks\" : {\"F24\" : {\"func\" : \"unknown\",\"id\" : \"F24\",\"trigger\" : \"click\",\"knownArgs\" : [\"27\"],\"requiredArgs\" : {}}}},\"F26\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F26\",\"size\" : 10,\"title\" : \"58\",\"color\" : \"\",\"layer\" : -2,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F28\" : {\"type\" : \"default\",\"id\" : \"F28\",\"text\" : \"{(0,1,1))\",\"title\" : \"\"}},\"callbacks\" : {\"F27\" : {\"func\" : \"unknown\",\"id\" : \"F27\",\"trigger\" : \"click\",\"knownArgs\" : [\"58\"],\"requiredArgs\" : {}}}},\"F29\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F29\",\"size\" : 10,\"title\" : \"89\",\"color\" : \"\",\"layer\" : -3,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F31\" : {\"type\" : \"default\",\"id\" : \"F31\",\"text\" : \"{(0,1,2))\",\"title\" : \"\"}},\"callbacks\" : {\"F30\" : {\"func\" : \"unknown\",\"id\" : \"F30\",\"trigger\" : \"click\",\"knownArgs\" : [\"89\"],\"requiredArgs\" : {}}}}},\"links\" : {\"F32\" : {\"id\" : \"F32\",\"source\" : \"F5\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F2\"},\"F33\" : {\"id\" : \"F33\",\"source\" : \"F8\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F5\"},\"F34\" : {\"id\" : \"F34\",\"source\" : \"F11\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F8\"},\"F35\" : {\"id\" : \"F35\",\"source\" : \"F14\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F23\"},\"F36\" : {\"id\" : \"F36\",\"source\" : \"F17\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F14\"},\"F37\" : {\"id\" : \"F37\",\"source\" : \"F17\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F26\"},\"F38\" : {\"id\" : \"F38\",\"source\" : \"F20\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F17\"},\"F39\" : {\"id\" : \"F39\",\"source\" : \"F20\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F29\"},\"F40\" : {\"id\" : \"F40\",\"source\" : \"F23\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F2\"},\"F41\" : {\"id\" : \"F41\",\"source\" : \"F26\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F5\"},\"F42\" : {\"id\" : \"F42\",\"source\" : \"F26\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F23\"},\"F43\" : {\"id\" : \"F43\",\"source\" : \"F29\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F8\"},\"F44\" : {\"id\" : \"F44\",\"source\" : \"F29\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F26\"}}},\"messages\" : {}}}"
    },
    "execution_count": 4,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "apery(NumericalSemigroup(10,51,27,31));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Drawing sons of numerical semigroups"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example shows how to draw the sons of a numerical semigroup in the tree of numerical semigroups. If we click on a node, then the sets of sons of that node are added to the canvas, and if the node is a leaf, a warning message is displayed.\n",
    "\n",
    "Passing the mouse over a node shows the set of minimal generators of the node."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( s ) ... end"
     ]
    },
    "execution_count": 5,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "sons:=function(s)\n",
    "    local gens, frb, desc, graphHasse, d, shpr, shp, canvas, sonsf, i, gn, lbl;\n",
    "    \n",
    "    \n",
    "    sonsf:=function(s,n)\n",
    "        local gens, frb, desc, d, shp, i, lbl, gn;\n",
    "\n",
    "        frb:=FrobeniusNumber(s);\n",
    "        gens:=Filtered(MinimalGenerators(s), x-> x>frb);\n",
    "        desc:=List(gens, g->RemoveMinimalGeneratorFromNumericalSemigroup(g,s));\n",
    "        gn:=Genus(s);\n",
    "        i:=0;\n",
    "        for d in desc do\n",
    "            i:=i+1;\n",
    "        lbl:=Concatenation(\"$\\\\langle\",JoinStringsWithSeparator(MinimalGenerators(d),\",\"),\"\\\\rangle$\");\n",
    "            shp:=Shape(ShapeType!.CIRCLE, lbl);\n",
    "            SetFrancyId(shp,lbl);\n",
    "            SetLayer(shp,Genus(d));\n",
    "            SetSize(shp,1);\n",
    "            Add(shp,Callback(sonsf,[d,shp]));\n",
    "            Add(shp,FrancyMessage(String(MinimalGenerators(d))));\n",
    "            Add(graphHasse,shp);\n",
    "            Add(graphHasse,Link(n,shp));\n",
    "        od;\n",
    "        if desc<>[] then \n",
    "            return Draw(canvas);\n",
    "        fi;\n",
    "        Add(canvas, FrancyMessage(FrancyMessageType.WARNING, \"This semigroup is a leaf\"));\n",
    "        return Draw(canvas);\n",
    "    end;\n",
    "    \n",
    "    frb:=FrobeniusNumber(s);\n",
    "    gens:=Filtered(MinimalGenerators(s), x-> x>frb);\n",
    "    desc:=List(gens, g->RemoveMinimalGeneratorFromNumericalSemigroup(g,s));\n",
    "    gn:=Genus(s);\n",
    "\n",
    "    graphHasse := Graph(GraphType.UNDIRECTED);\n",
    "    lbl:=Concatenation(\"$\\\\langle\",JoinStringsWithSeparator(MinimalGenerators(s),\",\"),\"\\\\rangle$\");\n",
    "    shpr:=Shape(ShapeType!.CIRCLE, lbl);\n",
    "    SetSize(shpr,1);\n",
    "    SetFrancyId(shpr,lbl);\n",
    "    Add(shpr,FrancyMessage(String(MinimalGenerators(s))));\n",
    "    SetLayer(shpr,Genus(s));\n",
    "    Add(graphHasse,shpr);\n",
    "    i:=0;\n",
    "    for d in desc do\n",
    "        i:=i+1;\n",
    "        lbl:=Concatenation(\"$\\\\langle\",JoinStringsWithSeparator(MinimalGenerators(d),\",\"),\"\\\\rangle$\");\n",
    "        shp:=Shape(ShapeType!.CIRCLE, lbl);\n",
    "        SetFrancyId(shp,lbl);\n",
    "        SetLayer(shp,Genus(d));\n",
    "        SetSize(shp,1);\n",
    "        Add(shp,Callback(sonsf,[d,shp]));\n",
    "        Add(shp,FrancyMessage(String(MinimalGenerators(d))));\n",
    "        Add(graphHasse,shp);\n",
    "        Add(graphHasse,Link(shpr,shp));\n",
    "    od;\n",
    "    canvas:=Canvas(\"Sons of a numerical semigroup\");\n",
    "    SetTexTypesetting(canvas, true);\n",
    "    Add(canvas,graphHasse);\n",
    "    return Draw(canvas);    \n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [{
    "data": {
     "application/vnd.francy+json": "{\"version\" : \"1.0.2\",\"mime\" : \"application\\/vnd.francy+json\",\"canvas\" : {\"width\" : 800,\"id\" : \"F57\",\"height\" : 600,\"title\" : \"Sons of a numerical semigroup\",\"zoomToFit\" : true,\"texTypesetting\" : true,\"menus\" : {},\"graph\" : {\"type\" : \"undirected\",\"id\" : \"F46\",\"simulation\" : true,\"collapsed\" : true,\"nodes\" : {\"$\\\\langle3,5,7\\\\rangle$\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"$\\\\langle3,5,7\\\\rangle$\",\"size\" : 1,\"title\" : \"$\\\\langle3,5,7\\\\rangle$\",\"color\" : \"\",\"layer\" : 3,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F48\" : {\"type\" : \"default\",\"id\" : \"F48\",\"text\" : \"[ 3, 5, 7 ]\",\"title\" : \"\"}},\"callbacks\" : {}},\"$\\\\langle3,7,8\\\\rangle$\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"$\\\\langle3,7,8\\\\rangle$\",\"size\" : 1,\"title\" : \"$\\\\langle3,7,8\\\\rangle$\",\"color\" : \"\",\"layer\" : 4,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F51\" : {\"type\" : \"default\",\"id\" : \"F51\",\"text\" : \"[ 3, 7, 8 ]\",\"title\" : \"\"}},\"callbacks\" : {\"F50\" : {\"func\" : \"unknown\",\"id\" : \"F50\",\"trigger\" : \"click\",\"knownArgs\" : [\"<object>\",\"<object>\"],\"requiredArgs\" : {}}}},\"$\\\\langle3,5\\\\rangle$\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"$\\\\langle3,5\\\\rangle$\",\"size\" : 1,\"title\" : \"$\\\\langle3,5\\\\rangle$\",\"color\" : \"\",\"layer\" : 4,\"parent\" : \"\",\"menus\" : {},\"messages\" : {\"F55\" : {\"type\" : \"default\",\"id\" : \"F55\",\"text\" : \"[ 3, 5 ]\",\"title\" : \"\"}},\"callbacks\" : {\"F54\" : {\"func\" : \"unknown\",\"id\" : \"F54\",\"trigger\" : \"click\",\"knownArgs\" : [\"<object>\",\"<object>\"],\"requiredArgs\" : {}}}}},\"links\" : {\"F52\" : {\"id\" : \"F52\",\"source\" : \"$\\\\langle3,5,7\\\\rangle$\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"$\\\\langle3,7,8\\\\rangle$\"},\"F56\" : {\"id\" : \"F56\",\"source\" : \"$\\\\langle3,5,7\\\\rangle$\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"$\\\\langle3,5\\\\rangle$\"}}},\"messages\" : {}}}"
    },
    "execution_count": 6,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "sons(NumericalSemigroup(3,5,7));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Tree of numerical semigroups\n",
    "\n",
    "Now we draw the sons of a numerical semigroup `s` in the tree of numerical semigroups up to level `l`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( s, l, generators ) ... end"
     ]
    },
    "execution_count": 7,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "sonstree:=function(s,l,generators)\n",
    "    local gens, frb, desc, graphTreee, d, shpr, shp, canvas, sonsf, lbl;\n",
    "\n",
    "\n",
    "    sonsf:=function(s,n,lv)\n",
    "        local gens, frb, desc, d, shp;\n",
    "        if lv=0 then\n",
    "            return ;\n",
    "        fi;\n",
    "        frb:=FrobeniusNumber(s);\n",
    "        gens:=Filtered(generators(s), x-> x>frb);\n",
    "        desc:=List(gens, g->RemoveMinimalGeneratorFromNumericalSemigroup(g,s));\n",
    "        for d in desc do\n",
    "            lbl:=Concatenation(\"$\\\\{\",JoinStringsWithSeparator(generators(d),\",\"),\"\\\\}$\");\n",
    "            shp:=Shape(ShapeType!.CIRCLE, lbl);\n",
    "            SetSize(shp,5);\n",
    "            Add(graphTreee,shp);\n",
    "            SetParentShape(shp,n);\n",
    "            sonsf(d,shp,lv-1);\n",
    "        od;\n",
    "        if desc<>[] then\n",
    "            return ;\n",
    "        fi;\n",
    "        #Add(canvas, FrancyMessage(FrancyMessageType.WARNING, \"This semigroup is a leaf\"));\n",
    "        return ;\n",
    "    end;\n",
    "\n",
    "    frb:=FrobeniusNumber(s);\n",
    "    gens:=Filtered(generators(s), x-> x>frb);\n",
    "    desc:=List(gens, g->RemoveMinimalGeneratorFromNumericalSemigroup(g,s));\n",
    "\n",
    "    graphTreee := Graph(GraphType.TREE);\n",
    "    SetCollapsed(graphTreee,false);\n",
    "    shpr:=Shape(ShapeType!.CIRCLE, \"S\");\n",
    "    SetSize(shpr,5);\n",
    "    Add(shpr,FrancyMessage(String(generators(s))));\n",
    "    Add(graphTreee,shpr);\n",
    "    canvas:=Canvas(\"Sons of a numerical semigroup\");\n",
    "    SetTexTypesetting(canvas, true);\n",
    "    Add(canvas,graphTreee);\n",
    "    sonsf(s,shpr,l);\n",
    "    return Draw(canvas);\n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Darker dots correspond either to leaves or to elements with highest genus. Blue nodes can be collapsed by clicking.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [{
    "data": {
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    "execution_count": 8,
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   "source": [
    "sonstree(NumericalSemigroup(1),4,MinimalGenerators);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And we can get also the tree of Arf numerical semigroups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
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   "source": [
    "sonstree(NumericalSemigroup(1),8,ArfCharactersOfArfNumericalSemigroup);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Oversemigroups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( s ) ... end"
     ]
    },
    "execution_count": 10,
    "metadata": {
     "text/plain": ""
    },
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   }],
   "source": [
    "oversemigroups:=function(s)\n",
    "    local ov, graphHasse, canvas,c,i,r,ovs,n,hasse,lbl;\n",
    "    \n",
    "    hasse:=function(rel)\n",
    "      local dom, out;\n",
    "      dom:=Flat(rel);\n",
    "      out:=Filtered(rel, p-> ForAny(dom, x->([p[1],x] in rel) and ([x,p[2]] in rel)));\n",
    "      return Difference(rel,out);\n",
    "    end;\n",
    "\n",
    "    ov:=OverSemigroupsNumericalSemigroup(s);\n",
    "    n:=Length(ov);\n",
    "    graphHasse := Graph(GraphType.UNDIRECTED);\n",
    "    #SetSimulation(graphHasse,true);\n",
    "    #SetDrag(graphHasse,true);\n",
    "    c:=Cartesian([1..n],[1..n]);\n",
    "    c:=Filtered(c, p-> p[2]<>p[1]);\n",
    "    c:=Filtered(c, p-> IsSubset(ov[p[1]],ov[p[2]]));\n",
    "    c:=hasse(c);\n",
    "    ovs:=[];\n",
    "    for i in [1..n] do\n",
    "    lbl:=Concatenation(\"$\\\\langle\",JoinStringsWithSeparator(MinimalGenerators(ov[i]),\",\"),\"\\\\rangle$\");\n",
    "\n",
    "\n",
    "        if IsIrreducible(ov[i]) then\n",
    "            ovs[i]:=Shape(ShapeType!.DIAMOND, lbl);\n",
    "        else\n",
    "            ovs[i]:=Shape(ShapeType!.CIRCLE, lbl);\n",
    "        fi;\n",
    "        SetLayer(ovs[i],Genus(ov[i]));\n",
    "    SetSize(ovs[i],2);\n",
    "    Add(graphHasse,ovs[i]);\n",
    "    od;\n",
    "    for r in c do\n",
    "        Add(graphHasse,Link(ovs[r[1]],ovs[r[2]]));\n",
    "    od;\n",
    "    canvas:=Canvas(\"Oversemigroups\");\n",
    "    SetTexTypesetting(canvas, true);\n",
    "    Add(canvas,graphHasse);\n",
    "    return Draw(canvas);    \n",
    "end;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A Hasse diagram of the set of oversemigroups of the given numerical semigroup is displayed. Irreducible numerical semigroups are drawn as diamonds. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "scrolled": false
   },
   "outputs": [{
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false,\"target\" : \"F144\"},\"F170\" : {\"id\" : \"F170\",\"source\" : \"F143\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F144\"}}},\"messages\" : {}}}"
    },
    "execution_count": 11,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "oversemigroups(NumericalSemigroup(6,7,9,11));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graphs associated to elements"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Eliahou graph of an element in a numerical semigroup is a graph whose vertices are the factorizations of the element, and to vertices are joined with an edge if they have common support."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( n, s ) ... end"
     ]
    },
    "execution_count": 12,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawEliahouGraph:=function(n,s)\n",
    "    local graph, canvas, f, fs, c, nf, i, p;\n",
    "    \n",
    "    f:=FactorizationsElementWRTNumericalSemigroup(n,s);\n",
    "    graph:=Graph(GraphType.UNDIRECTED);\n",
    "    #SetShowNeighbours(graph,true);\n",
    "    #SetSimulation(graph,true);\n",
    "    #SetDrag(graph,true);\n",
    "    nf:=Length(f);\n",
    "    fs:=[];\n",
    "    for i in [1..nf] do \n",
    "        fs[i]:=Shape(ShapeType!.CIRCLE, Concatenation(\"(\",JoinStringsWithSeparator(f[i],\",\"),\")\"));\n",
    "        SetLayer(fs[i],Sum(f[i]));\n",
    "        SetSize(fs[i],1);\n",
    "        Add(graph,fs[i]);\n",
    "    od;\n",
    "    c:=Cartesian([1..nf],[1..nf]);\n",
    "    c:=Filtered(c,p->p[1]<p[2] and f[p[1]]*f[p[2]]<>0);\n",
    "    for p in c do \n",
    "        Add(graph,Link(fs[p[1]],fs[p[2]]));\n",
    "    od;\n",
    "    canvas:=Canvas(\"Eliahou graph\");\n",
    "    Add(canvas,graph);\n",
    "    return Draw(canvas);\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "Numerical semigroup with 3 generators"
     ]
    },
    "execution_count": 13,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "s:=NumericalSemigroup(5,7,9);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "scrolled": false
   },
   "outputs": [{
    "data": {
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    },
    "execution_count": 14,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawEliahouGraph(49,s);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "[ 14, 25, 27 ]"
     ]
    },
    "execution_count": 15,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "BettiElements(s);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [{
    "data": {
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    },
    "execution_count": 16,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawEliahouGraph(55,s);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( n, s ) ... end"
     ]
    },
    "execution_count": 17,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawRosalesGraph:=function(n,s)\n",
    "    local graph, canvas, msg, msgs, c, nv, i, p;\n",
    "    \n",
    "    msg:=Filtered(MinimalGenerators(s), g->n-g in s);\n",
    "    graph:=Graph(GraphType.UNDIRECTED);\n",
    "    #SetSimulation(graph,true);\n",
    "    #SetDrag(graph,true);\n",
    "    nv:=Length(msg);\n",
    "    msgs:=[];\n",
    "    for i in [1..nv] do \n",
    "        msgs[i]:=Shape(ShapeType!.CIRCLE, String(msg[i]));\n",
    "        SetSize(msgs[i],1);\n",
    "        Add(graph,msgs[i]);\n",
    "    od;\n",
    "    c:=Cartesian([1..nv],[1..nv]);\n",
    "    c:=Filtered(c,p->p[1]<p[2] and n-(msg[p[1]]+msg[p[2]]) in s);\n",
    "    for p in c do \n",
    "        Add(graph,Link(msgs[p[1]],msgs[p[2]]));\n",
    "    od;\n",
    "    canvas:=Canvas(\"Rosales graph\");\n",
    "    Add(canvas,graph);\n",
    "    return Draw(canvas);\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "Numerical semigroup with 3 generators"
     ]
    },
    "execution_count": 18,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "s:=NumericalSemigroup(5,7,9);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [{
    "data": {
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    },
    "execution_count": 19,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawRosalesGraph(49,s);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [{
    "data": {
     "application/vnd.francy+json": "{\"version\" : \"1.0.2\",\"mime\" : \"application\\/vnd.francy+json\",\"canvas\" : {\"width\" : 800,\"id\" : \"F236\",\"height\" : 600,\"title\" : \"Rosales graph\",\"zoomToFit\" : true,\"texTypesetting\" : false,\"menus\" : {},\"graph\" : {\"type\" : \"undirected\",\"id\" : \"F231\",\"simulation\" : true,\"collapsed\" : true,\"nodes\" : {\"F232\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F232\",\"size\" : 1,\"title\" : \"3\",\"color\" : \"\",\"layer\" : 0,\"parent\" : \"\",\"menus\" : {},\"messages\" : {},\"callbacks\" : {}},\"F233\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F233\",\"size\" : 1,\"title\" : \"5\",\"color\" : \"\",\"layer\" : 0,\"parent\" : \"\",\"menus\" : {},\"messages\" : {},\"callbacks\" : {}},\"F234\" : {\"x\" : 0,\"y\" : 0,\"type\" : \"circle\",\"id\" : \"F234\",\"size\" : 1,\"title\" : \"7\",\"color\" : \"\",\"layer\" : 0,\"parent\" : \"\",\"menus\" : {},\"messages\" : {},\"callbacks\" : {}}},\"links\" : {\"F235\" : {\"id\" : \"F235\",\"source\" : \"F232\",\"length\" : 0,\"weight\" : 0,\"color\" : \"\",\"invisible\" : false,\"target\" : \"F234\"}}},\"messages\" : {}}}"
    },
    "execution_count": 20,
    "metadata": {
     "application/vnd.francy+json": {}
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawRosalesGraph(10,NumericalSemigroup(3,5,7));"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Graphs of factorizations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "function( f ) ... end"
     ]
    },
    "execution_count": 21,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "DrawFactorizationGraph:=function(f)\n",
    "    local graph, canvas, fs, c, nf, i, p, ln, distance, Kruskal, tv;\n",
    "\n",
    "    Kruskal := function(V, E)\n",
    "        local trees, needed, v, e, i,j, nv;\n",
    "\n",
    "        trees := List(V, v-> [v]);\n",
    "        needed := [];\n",
    "        nv:=Length(V);\n",
    "        for e in E do\n",
    "          i:=First([1..Length(trees)], k-> e[1] in trees[k]);\n",
    "          j:=First([1..Length(trees)], k-> e[2] in trees[k]);\n",
    "          if i<>j then\n",
    "            trees[i]:=Union(trees[i], trees[j]);\n",
    "            trees[j]:=[];\n",
    "            Add(needed,e);\n",
    "          fi;\n",
    "          if Length(needed)=nv-1 then\n",
    "            break;\n",
    "          fi;\n",
    "        od;\n",
    "        return needed;\n",
    "    end;\n",
    " \n",
    "    distance := function(a,b)\n",
    "        local   k,  gcd,  i;\n",
    "\n",
    "        k := Length(a);\n",
    "        if k <> Length(b) then\n",
    "            Error(\"The lengths of a and b are different.\\n\");\n",
    "        fi;\n",
    "\n",
    "\n",
    "        gcd := [];\n",
    "        for i in [1..k] do\n",
    "            Add(gcd, Minimum(a[i],b[i]));\n",
    "        od;\n",
    "        return(Maximum(Sum(a-gcd),Sum(b-gcd)));\n",
    "\n",
    "    end;\n",
    "\n",
    "    graph:=Graph(GraphType.UNDIRECTED);\n",
    "# SetSimulation(graph,true);\n",
    "#SetDrag(graph,true);\n",
    "#SetShowNeighbours(graph,true);\n",
    "    nf:=Length(f);\n",
    "    fs:=[];\n",
    "    for i in [1..nf] do \n",
    "        fs[i]:=Shape(ShapeType!.CIRCLE, Concatenation(\"(\",JoinStringsWithSeparator(f[i],\",\"),\")\"));\n",
    "        SetLayer(fs[i],Sum(f[i]));\n",
    "        SetSize(fs[i],1);\n",
    "        Add(graph,fs[i]);\n",
    "    od;\n",
    "    c:=Cartesian([1..nf],[1..nf]);\n",
    "    c:=Filtered(c,p->p[1]<p[2] and f[p[1]]*f[p[2]]<>0);\n",
    "    Sort(c,function(e,ee) return distance(f[e[1]],f[e[2]])<distance(f[ee[1]],f[ee[2]]); end);\n",
    "    tv:=Kruskal(f,List(c,p->[f[p[1]],f[p[2]]]));\n",
    "    for p in c do \n",
    "        ln:=Link(fs[p[1]],fs[p[2]]);\n",
    "        #SetWeight(ln, distance(f[p[1]],f[p[2]]));\n",
    "        SetTitle(ln, String(distance(f[p[1]],f[p[2]])));\n",
    "        if [f[p[1]],f[p[2]]] in tv then \n",
    "            SetColor(ln,\"red\");\n",
    "        fi;\n",
    "        Add(graph,ln);\n",
    "    od;\n",
    "    canvas:=Canvas(\"Factorizations graph\");\n",
    "    Add(canvas,graph);\n",
    "    return Draw(canvas);\n",
    "end;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [{
    "data": {
     "text/plain": [
      "[ [ 10, 2, 0 ], [ 5, 5, 0 ], [ 0, 8, 0 ], [ 11, 0, 1 ], [ 6, 3, 1 ], [ 1, 6, 1 ], [ 7, 1, 2 ], [ 2, 4, 2 ], [ 3, 2, 3 ], [ 4, 0, 4 ], [ 0, 1, 5 ] ]"
     ]
    },
    "execution_count": 22,
    "metadata": {
     "text/plain": ""
    },
    "output_type": "execute_result"
   }],
   "source": [
    "f:=FactorizationsElementWRTNumericalSemigroup(40,NumericalSemigroup(3,5,7));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "scrolled": false
   },
   "outputs": [{
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    },
    "execution_count": 23,
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   "source": [
    "DrawFactorizationGraph(f);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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