The parabolic map

In this paper we study certain aspects of Bruhat order on Coxeter groups. L et W be a Coxeter group, and let P denote the set of parabolic subgroups of W. We construct a map m: W x P --> W and examine its first properties. We prove the following characterization: w is the longest element of a parabol ic subgroup of W if and only if for any upsilon is an element of W, there e xists a unique maximal element (with respect to Bruhat order) less than or equal to both upsilon and w. (C) 1999 Academic Press.