Involutive divisions for effective involutive algorithms

Properties of involutive divisions on monomials are studied. A new method of involutive graphs is developed. The concept of complete global involutive division is introduced. A criterion of Noetherity of involutive divisions, a property of graphs of global involutive division, a test for completeness of global involutive division, a criterion of global involutive division are considered. A new series of involutive divisions is obtained by the process of completion. The properties of the divisions contained in the constructed series are studied. It is shown that the divisions from the series are better than the classical involutive divisions for involutive algorithms. The problem stated by Gao is solved: another series of involutive divisions is obtained. It is proved that all divisions of this series are continuous.