Let $E$ be a finite semigroup. According to N. Bourbaki (Algèbre I p. 121 exerc. 14 c), if $M$ and $M'$ are minimal right ideals in $E$, then they are isomorphic. I spent some time browsing through Clifford and Preston's monography "the algebraic theory of semigroups", also reading a few articles by Rees, Clifford and Dubreil, and found absolutely no mention of this fact anywhere. Maybe I haven't looked hard enough. If there is any semigroup specialist, could you tell me what you think?



Minimal right ideals in finite semigroup
This is in Clifford-Preston. First every minimal left (right) ideal is inside the minimal two-sided ideal $I$ (which is unique), see Exercise 13 on page 84. Second, the ideal $I$ is a simple semigroup (obvious, but is also in C-P). Third, by Sushkevich's theorem (Appendix A), all maximal subgroups of $I$ are isomorphic. By Theorem 1.27, every minimal left ideal $L$ in $I$ is a left group, i.e. the direct product of the maximal subgroup and a left zero semigroup (whose order does not depend of $L$). Hence all minimal left ideals are isomorphic. 


