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144 changes: 144 additions & 0 deletions notebooks/Congruence.ipynb
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{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Congruences\n",
"\n",
"Let $S$ be a semigroup and $\\rho \\subseteq S \\times S$. We say that is a $\\rho$ is a *left congruence* if $(sx,sy) \\in \\rho$ for all $(x,y) \\in \\rho$ and $s \\in S$. Similarly, we say that $\\rho$ is a right congruence if $(xs,ys) \\in \\rho$ for all $(x,y) \\in \\rho$ and $s \\in S$. If $\\rho$ is both a left and right congruence, then we say $\\rho$ is a congruence. \n",
"\n",
"A question we are interested in is, given a subset $\\rho \\subseteq S \\times S$, what is the \"smallest\" congruence that contains $(x,y)$. By smallest congruence, we mean this to be the intersection of every equivalence relation on $S$ containing $\\rho$. One way to do this is by using the *congruence by pairs* algorithm. \n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Python 3.8.2\r\n"
]
}
],
"source": [
"!python -V"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"from IPython.core.interactiveshell import InteractiveShell\n",
"InteractiveShell.ast_node_interactivity = \"all\""
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"from libsemigroups_cppyy import CongruenceByPairs, FroidurePin, Permutation, Transformation"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [],
"source": [
"S = FroidurePin(Transformation([1,2,3,0]), Transformation([1,0,2,3]))"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"hai\n"
]
},
{
"ename": "TypeError",
"evalue": "Could not instantiate make<libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >>:\n Failed to instantiate \"make<libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >>(int,libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >&)\"",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m<ipython-input-5-828c118cf6d8>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mC_right\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mCongruenceByPairs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m\"right\"\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m~/Desktop/libsemigroups_cppyy/libsemigroups_cppyy/cong_pair.py\u001b[0m in \u001b[0;36mCongruenceByPairs\u001b[0;34m(t, S)\u001b[0m\n\u001b[1;32m 30\u001b[0m \u001b[0;32mraise\u001b[0m \u001b[0mTypeError\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'Expected one of \"right\", \"left\", or \"twosided\"'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 31\u001b[0m \u001b[0mcp_type\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcppyy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgbl\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlibsemigroups\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mCongruenceByPairs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0melement_type\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 32\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mcppyy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mgbl\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mtype\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mS\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mpj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m: Could not instantiate make<libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >>:\n Failed to instantiate \"make<libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >>(int,libsemigroups::FroidurePin<libsemigroups::Transformation<unsigned char>,libsemigroups::FroidurePinTraits<libsemigroups::Transformation<unsigned char> > >&)\""
]
}
],
"source": [
"C_right = CongruenceByPairs(\"right\",S)\n"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Python 3.7.4\r\n"
]
}
],
"source": [
"!python -V\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import cppyy\n"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "WORK",
"language": "python",
"name": "work"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.8.2"
}
},
"nbformat": 4,
"nbformat_minor": 2
}