Right and left strongly locally testable semigroups and languages

The objective of this paper is to de ne and characterize the families of right (left) strongly locally testable languages (RSLT and LSLT ) and semigroups (RSLTS and LSLTS). Given k > 0, the membership of a w ord x to a RSLT (RSTk) language can be decided by exploring the segments of length k of x, as well as considering the order of appearance of those segments when we scan the pre xes of x. A semigroup S is RSTSk if tw o sequences ofS + having the same segments of length k and with the same order of appearance of those segments, de ne the same product in S. Both families do not correspond to each other. The family of RSLT (LSLT ) languages is not a variety of languages, while the family of RSLTS (LSLTS) semigroups is a variety of semigroups which corresponds to the intersection of the v arieties ofR-trivial (L-trivial) and righ t (left) locally testable semigroups. The varieties of languages which correspond to these varieties of semigroups will be called right (left) over testable and denoted asROTL (LOTL).