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I am studying the book "Representation Theory of Finite Monoids" by Benjamin Steinberg, I've tried searching online for the solution but I don't know how to search for it so google shows no results. In Lemma 1.7 the author says that is clear that:

Given an idempotent, $e$, of a finite monoid $M$ and $m_1, m_2 \in eMe$: $$ (eMe)m_1(eMe) = (eMe)m_2(eMe) \Rightarrow Mm_1M = Mm_2M $$ but I can't figure it out.

I can see how $(eMe)m_1(eMe) = (eMe)m_2(eMe) \Rightarrow eMm_1Me = eMm_2Me$ but then I get stuck.

Diogo Santos
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1 Answers1

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First note that if $m \in eMe$, then $eme = m$. Now, let $m_1, m_2 \in eMe$ and suppose that $$ (eMe)m_1(eMe) = (eMe)m_2(eMe) $$ Then $$ m_1 = eeem_1eee \in (eMe)m_1(eMe) = (eMe)m_2(eMe) \subseteq Mm_2M $$ and similarly $m_2 \in Mm_1M$. It follows that $Mm_1M = Mm_2M$.

J.-E. Pin
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