Rational languages and the Burnside problem (Q1079380)

This survey paper is mainly concerned with the following problem: give properties which characterize regularity of formal languages. Since regularity of a language is equivalent to the finiteness of its syntactic monoid, this problem is related to the Burnside problem for monoids: under the assumption that each cyclic submonoid of the monoid M is finite (i.e. M is periodic), when M is finite ? Periodicity is itself related to the classical language-theoretic property of pumping. We consider also bounded and commutative languages in this paper, which ends with similar questions on formal power series: rationality questions, boundedness of the support.