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Coclass theory for nilpotent semigroups via their associated algebras

Coclass theory has been a highly successful approach towards the investigation and classification of finite nilpotent groups. Here we suggest a similar approach for finite nilpotent semigroups. This differs from the group theory setting in that we additionally use certain algebras associated to the considered semigroups. We propose a series of conjectures on our suggested approach. If these become theorems, then this would reduce the classification of nilpotent semigroups of a fixed coclass to a finite calculation. Our conjectures are supported by the classification of nilpotent semigroups of coclass 0 and 1. Computational experiments suggest that the conjectures also hold for the nilpotent semigroups of coclass 2 and 3.