  The cut and join operations play important roles in tensor models in general.
We introduce a generalization of the cut operation associated with the higher
order variations and demonstrate how they generate operators in the
Aristotelian tensor model. We point out that, by successive choices of
appropriate variations, the cut operation generalized this way can generate
those operators which do not appear in the ring of the join operation,
providing a tool to enumerate the operators by a level by level analysis
recursively. We present a set of rules that control the emergence of such
operators.
