Given any finite alphabet A and positive integers m1,...,mk, congruences on A*, denoted by ¸∼(m1,...,mk) and related to a version of the Ehrenfeucht-Fraisse game, are defined. Level kof the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoidsA*/¸∼(m1,...,mk). A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For ℝAℝ≥2, a necessary and sufficient condition is given for A*/¸∼(m1,...,mk) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A*/∼(m1,...,mk) are discussed.