Does Green's $\mathcal{J}$-relation define a total order on the equivalence classes? In a semigroup $S$ we define $a\le_\mathcal{J} b$ iff $a=xby$ for some $x,y\in S^1$. Defining $a\equiv_\mathcal{J} b$ by $a\le_\mathcal{J} b$ and $b\le_\mathcal{J} a$ gives a partial order on $S/\equiv_\mathcal{J}$. Is this order always total?

It might be good to look at this order as the inclusion of principal two-sided ideals instead (as it's isomorphic). Perhaps the fact that you didn't look at it like this is why you didn't immediately answer the question yourself. By your nickname I assume you like rings. In rings, principal ideals in the ring-theoretic sense and principal ideals in the semigroup-theoretic sense are the exact same thing. In the commutative case, two-sided and one-sided principal ideals are the same thing too. Can you see examples of commutative rings you know in which the principal ideals don't form a chain? 
