@article{DELUCA1980300,
    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 S1 *[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/X2 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.},
    Author = {{De Luca}, Aldo and Restivo, Antonio},
    File = {A Characterization of Strictly Locally Testable Languages and Its Application to Subsemigroups of a Free Semigroup - 1-s2.0-S0019995880901801-main.pdf},
    ISSN = {0019-9958},
    Journal = {Information and Control},
    Number = {3},
    Pages = {300-319},
    Title = {A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup},
    URL = {https://www.sciencedirect.com/science/article/pii/S0019995880901801},
    Volume = {44},
    Year = {1980},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0019995880901801},
    bdsk-url-2 = {https://doi.org/10.1016/S0019-9958(80)90180-1},
    date-added = {2021-08-27 14:22:44 +0200},
    date-modified = {2021-08-27 14:22:44 +0200},
    doi = {10.1016/S0019-9958(80)90180-1}
}

@article{DELUCA1980300, 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 S1 *[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/X2 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.}, Author = {{De Luca}, Aldo and Restivo, Antonio}, File = {A Characterization of Strictly Locally Testable Languages and Its Application to Subsemigroups of a Free Semigroup - 1-s2.0-S0019995880901801-main.pdf}, ISSN = {0019-9958}, Journal = {Information and Control}, Number = {3}, Pages = {300-319}, Title = {A characterization of strictly locally testable languages and its application to subsemigroups of a free semigroup}, URL = {https://www.sciencedirect.com/science/article/pii/S0019995880901801}, Volume = {44}, Year = {1980}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0019995880901801}, bdsk-url-2 = {https://doi.org/10.1016/S0019-9958(80)90180-1}, date-added = {2021-08-27 14:22:44 +0200}, date-modified = {2021-08-27 14:22:44 +0200}, doi = {10.1016/S0019-9958(80)90180-1} }