A monoid is an [[Algebraic Structure|algebreic structure]] which contains an [[Underlying Set|underlying set]] $A$, an [[Associative Property|associative]] [[Binary Operation|binary operation]] $\circ$ (who is closed under $S$), and an [[Identity Property|identity]] of $\circ$ in $A$, denoted as $e$. Alternatively, a [[Monoid|monoid]] is a [[Semigroup]] whose [[Operation|operator]] has an [[Identity Property|identity]]. $\huge \circ : S\times S \to S \\ $ [[Universal Quantifier|For any]] $a,b,c\in S$, $\huge a\circ b \circ c = (a \circ b) = a \circ ( b \circ c) $ $\huge a \circ e $ We can denote the monoid by the three properties of it (the [[Underlying Set|underlying set]], [[Operation|operation]] and [[Identity Property|identity]]) as $M=[S,\circ, e]$. >[!info] If the [[Monoid]] has an [[Inverse Property|inverse]] for every element in its set, then that monoid also constitutes a [[Group]]