Equations and Monoid Varieties of Dot-Depth One and Two

Abstract Each level of the Straubing's hierarchy of aperiodic monoids can be parametrized in a natural way. This paper studies this parametrization for dot-depth one and two monoids. For level one, it is shown that the m th level is defined by a finite sequence of equations if and only if m = 1, 2 or 3. For level two, and for m ⩾ 1, a sequence of equations is given which is satisfied in the m th level and shown to ultimately define the 1st level.

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