path algebra and quiver

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Im studyng quivers with the text of Assem:enter image description here

My question is: In "the words on {$\alpha, \beta$}", refers it in particular to the free monoide kleene star? I dont uderstand the relation with the free monoid.

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Indeed, the set of all words on $\{\alpha, \beta\}$ is the free monoid over $\{\alpha, \beta\}$, usually denoted $\{\alpha, \beta\}^*$.