Small nonisomorphic semigroups
<style>
td {padding-left:3px;padding-right:3px;}
th {background-color:#f0f0f0;font-weight:normal;padding-left:3px;padding-right:3px;}
table {border-collapse:collapse;line-height:80%;}
</style>
Format as <select id="format" onchange="update()">
<option value="table" selected>Table</option>
<option value="html">HTML</option>
<option value="text">Text</option>
</select>
| Filter by
<b>Is subdirectly irreducible</b><input type="checkbox" name="si" id="siyes" onclick="document.getElementById('sino').checked=false;update()">yes
<input type="checkbox" name="si" id="sino" onclick="document.getElementById('siyes').checked=false;update()">no
<b>Is simple</b><input type="checkbox" name="simple" id="simpleyes" onclick="document.getElementById('simpleno').checked=false;update()">yes
<input type="checkbox" name="simple" id="simpleno" onclick="document.getElementById('simpleyes').checked=false;update()">no |
Find algebras of size <input type="text" id="minsize" size="1" maxlength="2" value="1"> to <input type="text" id="maxsize" size="1" maxlength="2" value="4"><input type="button" value="Search" onclick="update()"> | Counts: <span id="counts"></span>
<div id="algebras"></div>
<script>
function algebraToString(a,k) {
if (format=="html"){
var st="<div style=\"display:inline-block;border: 1px darkgray solid;\"><sup>"+num+"</sup>"+"<b>S</b><sub>"+a.length+","+k+"</sub><br>";
for (var i=0; i<a.length; i++) {
for (var j=0; j<a.length; j++) st = st+a[i][j]+(j==a.length-1?"":" ");
st = st+"<br>";
}
st = st+"</div> "; num++;
}else if (format=="table"){
var st="<table style=\"display:inline-block;border: 1px darkgray solid;\"><tr><td colspan=\""+(a.length+1)+"\"><sup>"+num+"</sup>"+"<b>S</b><sub>"+a.length+","+k+"</sub></td></tr><tr><th>·</th>";
for (var i=0; i<a.length; i++) st += "<th>"+i+"</th>";
st += "</tr>";
for (var i=0; i<a.length; i++) {
st += "<tr><th>"+i+"</th>";
for (var j=0; j<a.length; j++)
st = st+"<td>"+a[i][j]+"</td>";
st = st+"</tr>";
}
st = st+"</table> "; num++;
}else if (format=="text"){
var st="{name:\""+classname+"_{"+a.length+","+k+"}\", ";
st=st+"num:"+num+", op:{\"cdot\":[\n";
for (var i=0; i<a.length; i++) {
st = st+"[";
for (var j=0; j<a.length; j++)
st = st+a[i][j]+(j==a.length-1?"":",");
st = st+"]"+(i==a.length-1?"]}},":",")+"\n";
}
st = st+"\n"; num++;
}
return st;
}
function checkRelation(A,rel) {//rel is a partial binary relation
//Check that rel is transitive and compatible with the operations of A
var op;
var n = A.size;
for (var x=0; x<n; x++)
for (var y=0; y<n; y++)
if (rel[x][y]==1 && x!=y) {
for (var z=x+1; z<n; z++)
if (rel[y][z]==1)
if (rel[x][z]==0)
return false; //not transitive
for (var r in A.op) {
op = A.op[r];
if (op.length!=null)
if (op[0].length==null) {
if (rel[op[x]][op[y]]==0)
return false;
} else
for (var z=0; z<n; z++) {
if (rel[op[x][z]][op[y][z]]==0)
return false;
if (rel[op[z][x]][op[z][y]]==0)
return false;
}
}
}
return true;
}
function copyOf(arr) {
var a = new Array(arr.length);
for (var i=0; i<arr.length; i++) {
a[i] = new Array(arr[i].length);
for (var j=0; j<arr[i].length; j++) a[i][j] = arr[i][j];
}
return a;
}
function completeRelation(A,rel,i,j) {
// find next i,j where rel[i][j]=2=undefined; for each val=0 or 1
// set rel[i][j]=val, check transitivity and compatibility
// restore and return if no completetion,
// else call completeRelation(rel,i,j+1)
var ok = true;
while (ok && i<rel.length) {
while (j<rel.length && rel[i][j]!=2) j++;
if (j>=rel.length) {
j=0;
i++;
} else ok = false;
}
if (ok) congl[congl.length] = copyOf(rel);
else for (var val=0; val<2; val++){
rel[i][j] = val;
rel[j][i] = val;
ok = checkRelation(A,rel);
if (ok) completeRelation(A,rel,i,j+1);
rel[i][j] = 2;
rel[j][i] = 2;
}
}
function congruences(A) {
// A is a finite algebra (JavaScript object)
congl = [];
var rel = new Array(A.size);
for (var i=0; i<A.size; i++) {
rel[i] = new Array(A.size);
for (var j=0; j<A.size; j++)
if (i!=j) rel[i][j] = 2;
else rel[i][j]=1;
}
completeRelation(A,rel,0,0);
return congl;
}
function isSubrelation(R,S) { //assumes symmetry of relations
for (var i=0; i<R.length; i++)
for (var j=i+1; j<R.length; j++)
if (R[i][j]>S[i][j]) return false;
return true;
}
function conLatleq(cong) { //input list of 0-1-relations
var leq = new Array(cong.length);
for (var i=0; i<cong.length; i++) {
leq[i] = new Array(cong.length);
leq[i][i] = true;
for (var j=0; j<cong.length; j++)
if (i!=j) leq[i][j] = isSubrelation(cong[i],cong[j]);
}
return leq;
}
function leq2uppercovers(rel) {
var n = rel.length;
var uc = new Array(n);
for (var i=0; i<n; i++) {
uc[i] = [];
for (var j=0; j<n; j++)
if (rel[i][j] && i!=j) {
for (var k=0; k<n &&
!(rel[i][k] && i!=k && rel[k][j] && k!=j); k++);
if (k==n) uc[i][uc[i].length] = j;
}
}
return uc;
}
function congblock(co,i) {
var block = [];
for (var j=0; j<co.length; j++)
if (co[i][j]==1)
block[block.length] = j;
return block;
}
function cong2part(co) {
var part = [];
var flag = new Array(co.length);
for (var i=0; i<co.length; i++)
if (flag[i]==null) {
cb = congblock(co,i);
for (var j=0; j<cb.length; j++) flag[cb[j]]=true;
part[part.length] = cb;
}
return part;
}
function conLat(A) {
var conA = {/*name:"Con("+A.name+")",*/ rel:{}};
var cl = congruences(A);
conA.size = cl.length;
conA.eltname = new Array(conA.size);
for (var i=0; i<conA.size; i++) {
conA.eltname[i] = cong2part(cl[i]).join(" | ");
}
conA.rel.uc = leq2uppercovers(conLatleq(cl));
return conA;
}
function isSimple(A) {
return congruences(A).length==2;
}
function isSI(A) {
var conA = conLat(A);
return conA.rel.uc[0].length==1;
}
///////////////////////////////////////////////////////////////////////
classname="Sgrp";
counts=[];
filtcounts=[];
function initializeAlgebra(n) { // finite groupoid with n elements, {0,1,...,n-1}
var alg = new Array(n);
for (var i=0; i<n; i++) {
alg[i] = new Array(n);
for (var j=0; j<n; j++)
alg[i][j] = n; // all elements undefined (=n)
}
return alg;
}
function checkAxioms(a) {
var n,ok,x,y,z,xy,xyz;
n = a.length;
ok = true;
for (x=0; ok && x<n; x++) {
for (y=0; ok && y<n; y++) {
xy=a[x][y];
if (xy<n) {
// ok = !(a[y][x]<n && xy!=a[y][x]); // check commutativity
if (ok) {
for (z=0; ok && z<n; z++) {
xyz = a[xy][z];
if (xyz<n) // check associativity
ok = !(a[y][z]<n &&
a[x][a[y][z]]<n && xyz!=a[x][a[y][z]]);
}
}
}
}
}
return ok;
}
function checkPermutations(alg) {
var i,j,ok,p,q,qi,aqi,a,equal;
ok = true;
var n = alg.length;
for (p=0; ok && p<perms.length; p++) {//ok means alg <= qalg
q = perms[p];
qi = invperms[p];
equal = true;
for (i=0; equal && i<n; i++) {//equal means apij=qaij
for (j=0; equal && j<n; j++) {
aqi = alg[qi[i]][qi[j]];
a = alg[i][j];
equal = (aqi<n && a==q[aqi]);
if (!equal) ok = (aqi==n || a<=q[aqi]);
}
}
}
return ok;
}
function completeAlgebra(alg,i,j) {
// find next i,j where alg[i][j]=n=undefined; for each val=0..n-1
// set alg[i][j]=val, check axioms and permutations
// if ok, call completeAlg(alg,i,j+1) then restore and return
var ok = true;
var n = alg.length;
while (ok && i<n) {
while (j<n && alg[i][j]<n) j++;
if (j>=n) {
j = 0;
i++;
} else ok = false;
}
if (ok) {
counts[alg.length-1]++;
var add = (!simpleyes||!simpleno) && (!siyes||!sino);
if (add && (simpleyes||simpleno)) {
var simple = isSimple({size:alg.length,op:{cdot:alg}});
add = simpleyes && simple || simpleno && !simple;
}
if (add && (siyes||sino)) {
var si = isSI({size:alg.length,op:{cdot:alg}});
add = siyes && si || sino && !si;
}
if (add) {
algstr=algstr+algebraToString(alg,counts[alg.length-1]);
filtcounts[alg.length-1]++;
}
} else for (var val=0; val<n; val++) {
alg[i][j] = val;
ok = checkAxioms(alg);
if (ok) ok = checkPermutations(alg);
if (ok) completeAlgebra(alg,i,j+1);
alg[i][j] = n;
}
}
function nextPermutation(p) {
var q=[];
for (var j=p.length-2; j>=0 && p[j]>p[j+1]; j--);
if (j<0) return [];
for (var k=0; k<j; k++) q[k]=p[k];
for (var k=p.length-1; p[j]>p[k]; k--);
q[j]=p[k];
for (var i=p.length-1; i>j; i--) q[i]=p[j+p.length-i];
q[j+p.length-k]=p[j];
return q;
}
function permutations(m,n) { // return list of permutations of {m,...,n}
var perms = [];
perms[0]=[];
for (var i=0; i<=n-m; i++) perms[0][i]=m+i;
for (var i=0; perms[i].length>0; i++) {
perms[i+1]=nextPermutation(perms[i]);
}
perms.length=perms.length-1;
return perms;
}
function inversePermutation(p) {
var q=[];
for (var i=0; i<p.length; i++)
q[p[i]]=i;
return q;
}
function findAlgebras(size) {
counts[size-1]=0;
filtcounts[size-1]=0;
perms = permutations(0,size-1);
invperms = [];
for (var i=0; i<perms.length; i++) invperms[i]=inversePermutation(perms[i]);
alg = initializeAlgebra(size);
completeAlgebra(alg,0,0);
}
function findAlgebrasRange(minsize,maxsize) {
simpleyes = document.getElementById("simpleyes").checked;
simpleno = document.getElementById("simpleno").checked;
siyes = document.getElementById("siyes").checked;
sino = document.getElementById("sino").checked;
format = document.getElementById("format").value;
algstr=(format=="text"?classname+"=[\n":"");
num=0;
document.getElementById("algebras").innerHTML=(format=="text"?"<textarea rows=\"35\">Searching...</textarea>":"Searching...");
for (var i=minsize; i<=maxsize; i++)
findAlgebras(i);
document.getElementById("algebras").innerHTML=(format=="text"?"<textarea rows=\"35\">"+algstr.slice(0,algstr.length-3)+"];\n</textarea>":algstr);
document.getElementById("counts").innerHTML="<a href=\"http://www.research.att.com/~njas/sequences/?q="+filtcounts.join(",+")+"&sort=0&fmt=0&language=english&go=Search\"><u><b>"+filtcounts.join(", ")+"</b></u></a>";
}
function update(){
findAlgebrasRange(eval(document.getElementById('minsize').value), eval(document.getElementById('maxsize').value));
}
update();
</script>
