A polynomial time algorithm for local testability and its level

A locally testable semigroup S is a semigroup with the property that for some non-negative integer k, called the order or level of local testability, two words u and v in some set of generators for S are equal in the semigroup if (1) the prefix and suffix of the words of length k coincide, and (2) the set of intermediate substrings of length k of the words coincide. The local testability problem for semigroups is, given a finite semigroup, to decide, if the semigroup is locally testable or not. Recently, we introduced a polynomial time algorithm for the local testability problem and to find the level of local testability for semigroups based on our previous description of identities of k-testable semigroups and the structure of locally testable semigroups. The first part of the algorithm we introduce solves the local testability problem. The second part of the algorithm finds the order of local testability of a semigroup. The algorithm is of order n², where n is the order of the semigroup.