Monoids in General

Definition

The general theory on monoids. A monoid is a tuple \( M = (S,\star) \) of a set \( S \) and a binary operator \( \star \) for which the following two axioms hold:

1. Associativity: For every \( a,b,c \in G \) we have that $$a \star (b \star c) = (a \star b) \star c$$
2. The existence of an identity element: We have one unique element \( 1 \in M \) such that for every \( a \in M \) we have $$1 \star a = a \star 1 = a$$

The second axiom ensures that no monoid is empty. If we furthermore for every \( a,b \in M \) have that $$ a \star b = b \star a $$ we say that \( M \) is abelian or that \( M \) is commutative.

Examples

• The set \( \mathbb{N} = \{0, 1, 2, ...\} \) with the operator + forms a monoid. Call it \( M = (\mathbb{N},+) \). We have that the operator is associative. We have that \( a + 0 = 0 + a = a \) for all \( a \in M \).
• The set of all strings \( S \) forms a monoid with concatenation as the operator. That is \( M = (S,*) \). We have that concatenation is associative. Let \( \varepsilon \) be the empty string. We have that \( \varepsilon * s = s * \varepsilon = s \) for every \( s \in S \).

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