The Deletion Order and Coxeter Groups

The deletion order of a finitely generated Coxeter group W is a total order on the elements which, as is proved, is a refinement of the Bruhat order. This order is applied in 8 to construct Elnitsky tilings for any finite Coxeter group. Employing the deletion order, a corresponding normal form of an element w of W is defined which is shown to be the same as the normal form of w using right to left lexicographic ordering. Further results on the deletion order are obtained relating to the property of being Artinian and, when W is finite, its interplay with the longest element of W.