  In this paper, in addition to the earlier introduced involutive divisions, we
consider a new class of divisions induced by admissible monomial orderings. We
prove that these divisions are noetherian and constructive. Thereby each of
them allows one to compute an involutive Groebner basis of a polynomial ideal
by sequentially examining multiplicative reductions of nonmultiplicative
prolongations. We study dependence of involutive algorithms on the completion
ordering. Based on properties of particular involutive divisions two
computational optimizations are suggested. One of them consists in a special
choice of the completion ordering. Another optimization is related to
recomputing multiplicative and nonmultiplicative variables in the course of the
algorithm.
