Run languages

The family ℛ of run languages is a generalization of the family of finite/cofinite languages. For an alphabet A, the family W⋇ of “⋇words” consists of finite products of factors each of which is of the form {a} or a+, a∃A, and ℛ is the Boolean closure of W⋇. It is shown that each run language or its complement is a finite union of ⋇words, and that ℛ is also the Boolean closure of W+, the set of “words” over {a+:a∃A}. Lastly we prove that ℛ is contained in the family γ1 of languages whose syntactic monoids are J-trivial, and that for a two-letter alphabet ℛ = γ1