Games, Equations and Dot-Depth Two Monoids

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 k of the Straubing hierarchy of aperiodic monoids can be characterized in terms of the monoids A*/~(m1, ... mk). A natural subhierarchy of level 2 and equation systems satisfied in the corresponding varieties of monoids are defined. For A ≥ 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. Article: