Abstract
Two classes of involutive divisions are considered: pairwise and sliced ones. Pairwise involutive divisions are convenient for the calculation of multiplicative and nonmultiplicative variables when new monomials are added. Directed sliced partitions are convenient for finding involutive divisors of an arbitrary monomial. It is proved that the only homothetic division that is simultaneously pairwise and directed sliced is the Janet involutive division. This shows that the Janet division is optimal for the calculation of multiplicative variables and for the fast search of a divisor.
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