Maps into Dynkin diagrams arising from regular monoids

Abstract

It has been shown by one of the authors that the system of idempotents of monoids on a group G of Lie type with Dynkin diagram T can be classified by the following data: a partially ordered set 2/ with maximum element 1 and a map A: ^ -> 2 r with A(l) = V and with the property that for all J\, J 2 , J} €%f with J\ < J 2 < J3, any connected component of X(J 2 ) is contained in either X(J\) or A(J$). In this paper we show that X comes from a regular monoid if and only if the following conditions are satisfied:(1) % is a A-semilattice;If J\,J 2 e 2^with J\ < J 2 and if A" is a two element discrete subset of k{J\) U X(J 2 ), then X C X(J) for some J e % with /, < J < J 2 .1980 Mathematics subject classification {Amer. Math. Soc.) (1985 Revision): 20 G 99, 20 M 17. By a Coxeter group W = (W, F) is meant a group W generated by a subset F of elements of order 2, such that W has a presentation by the relations ( a e)m{o,e) = i, for a, (9 e T. We assume that the rank \T\ < 00. If a, 6 e F, define a -6 if m = m{a,d) > 3. In this way T becomes a graph, called the Coxeter graph of W. Note that a, 6 are not adjacent in the graph if and only if ad = 0a. It is customary to write a -6 to mean m -3, a -6 to mean w = 4 and <r = 0 to mean m = 6. The possible graphs for finite H 7 were determined by Coxeter (see [8]).