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In a monoid, does $x \cdot y=e$ imply $y \cdot x=e$?

A monoid is a set $S$ together with a binary operation $\cdot:S \times S \rightarrow S$ such that: The binary operation $\cdot$ is associative, that is, $(a\cdot b) \cdot c=a\cdot (b \cdot c)$ for ...