A Kleene algebra $(A, \cdot, +, ^*, 0, 1)$ is an idempotent semiring $(A, \cdot, +, 0, 1)$ with an additional (right-associative) unary operator $^*$ , called the Kleene star, which satisfies

For a given alphabet $\Sigma$ , the set of all languages over $\Sigma$ , as well as the set of all regular languages over $\Sigma$ , are examples of Kleene algebras. Similarly, sets of regular expressions (regular sets) over $\Sigma$ are a form (or close variant) of a Kleene algebra: let $A$ be the set of all regular sets over a set $\Sigma$ of alphabets. Then $A$ is a Kleene algebra if we identify $\varnothing$ as $0$ , the singleton containing the empty string $\lambda$ as $1$ , concatenation operation as $\cdot$ , the union operation as $+$ , and the Kleene star operation as $^*$ . For example, let $a$ be a set of regular expression, then $$a^*=\lbrace \lambda \rbrace \cup a \cup a^2 \cup \cdots \cup a^n \cup \cdots,$$ so that $$aa^*=a \cup a^2 \cup \cdots \cup a^n \cup \cdots.$$ Adding $1$ on both sides and we have $$1+aa^*=\lbrace \lambda \rbrace \cup aa^*=\lbrace \lambda \rbrace \cup a \cup a^2 \cup \cdots \cup a^n \cup \cdots = a^*.$$ The other conditions are checked similarly.

Remark. There is another notion of a Kleene algebra, which arises from lattices. For more detail, see here.