Over testable languages

Strongly locally testable semigroups with commuting idempotents and related languages

“…The following combinatorial description of locally and piecewise testable languages is given in [19].…”

“…Locally and piecewise testable languages are strongly locally testable [19] and hence, since the family of strongly locally testable languages does not form a variety of languages, they constitute a strict subclass of the strongly locally testable languages.…”

“…A semigroup S is locally testable if whenever two words over the alphabet S have the same factors of a fixed length k, then the products of the letters of these words are equal. The variety of languages associated to the variety of strongly locally testable semigroups is the class of languages that are both locally testable and piecewise testable [19].…”

“…a n B * n with various conditions on the letters a i and the subsets B i of A (e.g. piecewise testable languages (Simon [20]), R-trivial languages (Eilenberg [11]), doth-depth two languages (Pin and Straubing [15]), Ash, Hall and Pin's result on commuting idempotents [8], over testable languages [19], etc. ).…”

“…S is strongly locally testable if whenever two words over the alphabet S have the same factors of a fixed length k, then the products of the letters of these words are equal. We had previously proved [19] that the syntactic semigroup of a rational language L is strongly locally testable if and only if L is both locally and piecewise testable. We characterize in this paper the variety of strongly locally testable semigroups with commuting idempotents and, using the theory of implicit operations on a variety of semigroups, we derive an elementary combinatorial description of the related variety of languages.…”