Abstract
A notion of ω-rigid sets for a finite monoid is introduced. We prove that a finite monoid M is the Arnold's syntactic monoid of some rational ω-language (ω-syntactic for short) if and only if there exists an ω-rigid set for M. This property is shown to be decidable for the finite monoids. Relationship between the family of ω-syntactic monoids and that of ∗-syntactic monoids (i.e. the syntactic monoids of rational languages of finite words) is established.
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