A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup

Abstract

A syntactic characterization of strictly locally testable languages is given by means of the concept of constant . If S is a semigroup and X a subset of S , an element c of S is called constant for X if for all p, q, r, s ε S 1 *[ pcq, rcs εX ⇒ pcs ε X ]. The main result of the paper states that a recognizable subset X of a free semigroup A + is strictly locally testable if and only if all the idempotents of the syntactic semigroup S ( X ) of X are constants for X′ = XΦ , where Φ : A + → S ( X ) is the syntactic morphism. By this result some remarkable consequences are derived for recognizable subsemigroups of A + . In particular we prove that if X is a recognizable free subsemigroup of A + and Y = X/X 2 its base then the following conditions are equivalent : (1) X is strictly locally testable. (2) X is locally testable. (3) X is locally parsable and Y is strictly locally testable. (4) X has a bounded synchronization delay and Y is strictly locally testable (5) A positive integer k exists such that all the elements of A + whose length is greater than or equal to k , are constants for X . (6) For all the idempotents e of the syntactic semigroup S ( X ) of X , eS ( X ) e ⊆ e , 0.