3
$\begingroup$

Kindly, I am asking to light my mind by some leading hints:

Can minimal and maximal ideals in a finite non-commutative semigroup $S$ coincide?

Thanks for the time!

  • 0
    Is your definition of non-commutative not-necessarily-commutative? – 2017-02-25
  • 0
    Hmmm you look familiar in someway, Resident Dementor? And your profile looks awfully familiar! 8-) It really is good to see you still around! – 2017-04-30
  • 0
    @amWhy: Oh yes! Thanks my friend. I am still Babak. :-) – 2017-05-02

1 Answers 1

2

If you just consider ideals, every finite non-commutative group is a solution to your question. If you consider proper ideals, the minimal examples are the monoid $\{1, a, b\}$ with $aa = ba = a$ and $ab = bb = b$ and its dual version, the monoid $\{1, a, b\}$ with $aa = ab = a$ and $ba = bb = b$.

  • 0
    Thanks for your idea. Honestly, before asking this question, I had been thinking this question in an finite Archimedean semigroup. Do you think it is true in such that semigroup? Thanks again – 2017-03-11
  • 0
    No. Take the semigroup $\{a, a^2, a^3\}$ in which $a^3 = a^4$. Then $\{a^3\}$ is the unique minimal ideal, but $\{a^2, a^3\}$ is a proper maximal ideal. – 2017-03-11