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!
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Mikasa
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Is your definition of non-commutative not-necessarily-commutative? – Jacob Wakem Feb 25 '17 at 04:15
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Hmmm you look familiar in someway, Resident Dementor? And your profile looks awfully familiar! 8-) It really is good to see you still around! – amWhy Apr 30 '17 at 18:05
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@amWhy: Oh yes! Thanks my friend. I am still Babak. :-) – Mikasa May 02 '17 at 09:20
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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 $ba = bb = b$.
J.-E. Pin
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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 – Mikasa Mar 11 '17 at 04:35
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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. – J.-E. Pin Mar 11 '17 at 09:41