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Unfolding all definitions, a monoid $(X,\ast)$ fulfills the following properties:

Axiom: Axioms of Monoid

1. closure: $x \ast y\in X$ for all $x,y\in X$.
2. associativity: \((x\ast y)\ast z=x\ast (y\ast z)\,\).
3. existence of neutral element: There is an element $e\in X$ with $e\ast x=x\ast e=x$ for all $x\in X$.

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References

Bibliography

1. Knauer Ulrich: "Diskrete Strukturen - kurz gefasst", Spektrum Akademischer Verlag, 2001
2. Lang, Serge: "Algebra - Graduate Texts in Mathematics", Springer, 2002, 3rd Edition