Omega Language - Operations

Operations

Some common operations defined on ω-languages are:

  • Intersection and union. Given ω-languages L and M, both L ∪ M and L ∩ M are ω-languages.
  • Left catenation. Let L be an ω-language, and K be a language of finite words only. Then K can be catenated on the left only to L to yield the new ω-language KL.
  • Omega (infinite iteration). As the notation hints, the operation ω is the infinite version of the Kleene star operator on finite-length languages. Given a formal language L, Lω is the ω-language of all infinite sequence of words from L; in the functional view, of all functions →L.
  • Prefixes. Let w be an ω-word. Then the formal language Pref(w) contains every finite prefix of w.
  • Limit. Given a finite-length language L, an ω-word w is in the limit of L if and only if Pref(w) ∩ L is an infinite set. In other words, for an arbitrarily large natural number n, it is always possible to choose some word in L, whose length is greater than n, and which is a prefix of w. The limit operation on L can be written Lδ or .

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