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Romania District MO 2022 Grade 12 P1

Source: Romania District MO 2022 Grade 12

March 27, 2022
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Problem Statement

Let ee be the identity of monoid (M,⋅)(M,\cdot) and a∈Ma\in M an invertible element. Prove that
[*]The set Ma:={x∈M:ax2a=e}M_a:=\{x\in M:ax^2a=e\} is nonempty; [*]If b∈Mab\in M_a is invertible, then b−1∈Mab^{-1}\in M_a if and only if a4=ea^4=e; [*]If (Ma,⋅)(M_a,\cdot) is a monoid, then x2=ex^2=e for all x∈Ma.x\in M_a.
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