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{
"cells": [
{
"cell_type": "markdown",
"id": "14e63719-e7e0-4505-80cb-66054ce61573",
"metadata": {},
"source": [
"## smallantimagmas\n",
"\n",
"We call a magma $M = \\langle M, \\ast \\rangle$ **anti-associative** if for all $a,b,c \\in M$, we have:\n",
"\n",
"$$\n",
"\\left( a \\ast b \\right) \\ast c \\not= a \\ast \\left( b \\ast c \\right).\n",
"$$\n",
"\n",
"---"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "fe3a2e44-1a11-4fcc-a61d-c96718e040b6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 4,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"LoadPackage(\"smallantimagmas\");"
]
},
{
"cell_type": "markdown",
"id": "ab5189ac-40c8-42a6-886e-9c5c7655eecd",
"metadata": {},
"source": [
"## List non-isomorphic anti-associative groupoids\n",
"\n",
"Lets list non-isomorphic groupoids of order $3$."
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "f96ef9d6-c2d5-4f3c-a887-76cfe53d6f35",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"[ <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators>, <magma with 3 generators> ]"
]
},
"execution_count": 3,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"Ms := AllSmallAntimagmas(3);"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "eaeec858-bd6e-4e22-8728-54928b4072c6",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<magma with 3 generators>"
]
},
"execution_count": 5,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"OneSmallAntimagma(3);"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "d3cdfe06-fa63-47eb-a6a5-383fb7b6f921",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<magma with 3 generators>"
]
},
"execution_count": 7,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"SmallAntimagma(3, 1);"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "c745a474-db2b-44dc-9e2f-7b7c71427ab4",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<magma with 2 generators>"
]
},
"execution_count": 8,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"M := SmallAntimagma(2, 1);"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "29d1438d-731e-44d6-aeda-39d75c68381f",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"true"
]
},
"execution_count": 9,
"metadata": {
"text/plain": ""
},
"output_type": "execute_result"
}
],
"source": [
"IsLeftCancellative(M);"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "904afd6b-3deb-401f-928f-91f0605c7aff",
"metadata": {},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "GAP 4",
"language": "gap",
"name": "gap-4"
},
"language_info": {
"codemirror_mode": "gap",
"file_extension": ".g",
"mimetype": "text/x-gap",
"name": "GAP 4",
"nbconvert_exporter": "",
"pygments_lexer": "gap",
"version": "4.14.0"
}
},
"nbformat": 4,
"nbformat_minor": 5
}
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