function b = is_loctest(S)
% is the monoid S locally testable?
% NOTE (IMPORTANT!!): use [S,STR]=dfsa2monoid_str(M,1)
[n,n] = size(S);
Id = find_idempotents(S);
b = 1;
for e = Id(:)'
  for s=1:n
    ese = prod3(S,e,s,e);
    if S(ese,ese)~=ese
      b = 0;
      return
    end
    for t=1:n
      ete = prod3(S,e,t,e);
      if S(ese,ete)~=S(ete,ese)
        b = 0;
        return
      end
    end
  end
end
return

function Id = find_idempotents(S)
Id = [];
[n,n] = size(S);
for i=2:n
  if S(i,i)==i
    Id(end+1) = i;
  end
end
return

function p = prod3(S,a,b,c)
% p = a*b*c in S
p = S(S(a,b),c);
return

function Si = Sab(S,a)
[n,n] = size(S);
Si = [];
for i=1:n
  for j=1:n
    Si(end+1) = S(S(i,a),j);
  end
end
return