Identities of locally testable semigroups

Let S be a semigroup of words over an alphabet Σ. Suppose that every two words a and c over Σ are equal in S if (1) the sets of subwords of length k of the words a and c coincide and are non-empty. (2) the prefix (suffix) of a of length k - 1 is equal to the prefix (suffix) of c. Then S is called k-testable. A semigroup is locally testable if it is k-testable for some k > 0. We present a finite basis of identities of the variety of k-testable semigroups. The structure of k-testable semigroup is studied. Necessary and sufficient conditions for local testability will be given. A solution to one problem from the survey of Shevrin and Sukhanov (1985) will be presented.