Abstract
Given any finite alphabet A and positive integers m 1,…, m k , congruences on A ∗, denoted by ~( m 1,…, m k ) 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 ∗/~( m 1,…, m k . 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 ∗/~( m 1,…, m k ) to be of dot-depth exactly 2. Upper and lower bounds on the dot-depth of the A ∗/~( m 1,…, m k ) are discussed.
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