Interval temporal logics provide a general frame-work for temporal representation and reasoning, where classical (point-based) linear temporal logics can be recovered as special cases. In this paper, we study the effects of the addition of an equivalence relation ~ to one of the most representative interval temporal logics, namely, the logic ?????? ¯ of Allen's relations meets, begun by, and begins. We first prove that the satisfiability problem for the resulting logic ?????? ¯ ~ remains decidable over finite linear orders, but it becomes nonprimitive recursive, while decidability is lost over N. We also show that decidability over N can be recovered by restricting to a suitable subset of models. Then, we show that ?????? ¯ ~ is expressive enough to define ????-regular languages, thus establishing a promising connection between interval temporal logics and extended ??-regular languages.