Positive closure of a formal language ($L^{+}$)

The positive closure of a formal language $L$ is the language of all concatenations of one or more words from $L$:Here $L^n$ is $n$-fold concatenation of formal languages, while the Kleene star also permits $n=0$, whose contribution is the empty word $\epsilon$. Thus $L^{\ast }=L^{+}\cup \{\epsilon \}$.

For $L=\{ab\}$, positive closure is $\{ab,abab,ababab,\dots \}$. If $\epsilon \in L$, then $\epsilon \in L^{+}$ as well: positive counts chosen words, not their total character length. For the empty language, ${\varnothing }^{+}=\varnothing$ but ${\varnothing }^{\ast }=\{\epsilon \}$. This operation preserves regular languages and is one of the closure requirements in an abstract family of languages.