GAP Code for Regularity Analysis of Small Semigroups.txt

# Load necessary packages
LoadPackage("semigroups");
LoadPackage("smallsemi");

# Define the GLSG regularity check function
IsRegularGLSG := function(S)
    local elements, multTable, n, i, j, k, NS, NR, NC, Q, Q_values;
    elements := Elements(S);
    n := Size(elements);
    multTable := List(elements, x -> List(elements, y -> Position(elements, x * y)));
    Q_values := [];
    
    # Compute Q(i,j) for regularity check
    for i in [1..n] do
        for j in [1..n] do
            k := multTable[i][j];
            # Count all factorizations of sk
            NS := Sum([1..n], p -> Size(Filtered([1..n], q -> multTable[p][q] = k)));
            # Count NR(si, sk)
            NR := Size(Filtered([1..n], q -> q<>j and multTable[i][q] = k));
            # Count NC(sj, sk)
            NC := Size(Filtered([1..n], p -> p<>i and multTable[p][j] = k));
            # Calculate Q
            Q := NS - 2 * NR - 2 * NC;
            Add(Q_values, Q);
        od;
    od;
    
    # Check if all Q values are equal
    return Size(Set(Q_values)) = 1;
end;

# Main analysis function
RunGLSGAnalysis := function()
    local n, allSems, total, regularCount;
    for n in [1..4] do
        allSems := AllSmallSemigroups(n);
        total := Size(allSems);
        regularCount := Size(Filtered(allSems, IsRegularGLSG));
        Print("Order ", n, ": ", total, " semigroups, ",
              regularCount, " with regular GLSGs (",
              String(100.0 * regularCount / total), "%)\n");
    od;
end;

# Execute the analysis
RunGLSGAnalysis();