Kleene star
In mathematical logic and computer science, the Kleene star (or Kleene operator or Kleene closure) is a unary operation, either on sets of strings or on sets of symbols or characters. In mathematics it is more commonly known as the free monoid construction. The application of the Kleene star to a set formula_1 is written as "formula_2". It is widely used for regular expressions, which is the context in which it was introduced by Stephen Kleene to characterize certain automata, where it means "zero or more repetitions". The set "formula_2" can also be described as the set containing  the empty string and all finite-length strings that can be generated by concatenating arbitrary elements of formula_1, allowing the use of the same element multiple times. If formula_1 is either the empty set ∅ or the singleton set formula_14, then formula_15; if formula_1 is any other finite set or countably infinite set, then "formula_2" is a countably infinite set. As a consequence, each formal language over a finite or countably infinite alphabet formula_18 is countable, since it is a subset of the countably infinite set formula_19. The operators are used in rewrite rules for generative grammars. Definition and notation . Given a set formula_1 define and define recursively the set If formula_1 is a formal language, then formula_26, the formula_27-th power of the set formula_1, is a shorthand for the concatenation of set formula_1 with itself formula_27 times. That is, "formula_26" can be understood to be the set of all strings that can be represented as the concatenation of formula_27 strings in formula_1. The definition of Kleene star on formula_1 is This means that the Kleene star operator is an idempotent unary operator: formula_36 for any set formula_1 of strings or characters, as formula_38 for every formula_39. Kleene plus . In some formal language studies, (e.g. AFL theory) a variation on the Kleene star operation called the "Kleene plus" is used. The Kleene plus omits the formula_40 term in the above union. In other words, the Kleene plus on formula_1 is or Examples. Example of Kleene star applied to set of strings: Example of Kleene plus applied to set of characters: Kleene star applied to the same character set: Example of Kleene star applied to the empty set: Example of Kleene plus applied to the empty set: where concatenation is an associative and noncommutative product. Example of Kleene plus and  Kleene star applied to the singleton set containing the empty string: Generalization. Strings form a monoid with concatenation as the binary operation and ε the identity element. The Kleene star is defined for any monoid, not just strings. More precisely, let ("M", ⋅) be a monoid, and "S" ⊆ "M". Then "S" is the smallest submonoid of "M" containing "S"; that is, "S" contains the neutral element of "M", the set "S", and is such that if "x","y" ∈ "S", then "x"⋅"y" ∈ "S". Furthermore, the Kleene star is generalized by including the *-operation (and the union) in the algebraic  structure itself by the notion of complete star semiring.