Transitive maps from posets to Dynkin diagrams (Q915881)

A map \(\lambda\) : \(P\to 2^{\Gamma}\) from a poset with maximum 1 to the set of induced subgraphs of a finite graph \(\Gamma\) is said to be transitive if \(\lambda (1)=\Gamma\) and, for a,b,c\(\in P\) such that \(a<b<c\), each connected component of \(\lambda\) (b) is either contained in \(\lambda\) (a) or contained in \(\lambda\) (c). \textit{M. Putcha} showed that transitive maps arise from the characterization of the system of idempotents of algebraic monoids on groups with BN-pairs, \(\Gamma\) being then the corresponding Dynkin diagram and P the lattice of principal ideals of the monoid [J. Algebra 120, 139-169 (1989; Zbl 0683.20051)]. The map \(\lambda\) is irreducible if, for any transitive map \(\psi\) : \(P_ 1\to 2^{\Gamma}\) and surjective order preserving mapping \(\theta\) : \(P\to P_ 1\) such that \(\lambda =\psi \circ \theta\), \(\theta\) is an isomorphism of posets. The authors construct a ``universal'' irreducible transitive map for a finite graph \(\Gamma\), showing how to obtain from it any other (irreducible) transitive map for \(\Gamma\). Examples from the theory of algebraic monoids are given to illustrate these results.