Language concatenation behaves as a “multiplicative” operator — _ is a “zero” and {_} is an “identity”. In this sense we define powers of a language.
For L ____* define L0 = {_}, and inductively for nŤ0 Ln+1 = Ln•L. Language powers satisify the usual laws of exponents — Ln•Lm = Ln+m.
In terms of these powers we define two other language operations. The Kleene closure (or star) of L ___* is
L* =
_
n = 0
Ln
_
And the positive closure is
L+ =
_
n = 1
_
Ln
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