Idempotencies received a great deal of interest through a problem stated by Burnside in 1902: Is every group, which satisfies the identity x^r=1 and has a finite set of generators, finite? In the context of Formal Languages, the derived problem of non-counting classes, also called Brzozowski's Problem, remained open for over 30 years. We treat a variant of this, where the relations in question can be applied only in one direction. That is, they always increase or decrease a word's length. The main motivation for this came from the field of DNA computation. The operation of duplication, which plays a role there, is just one particular case of such a relation. In contrast to non-counting classes, here many of the arising languages are not regular but rather complex. Thus many interesting problems remain to be solved.

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