Idempotent lattices, Renner monoids and cross section lattices of the special orthogonal algebraic monoids

Let MSOn (n is even) be the special orthogonal algebraic monoid, T a maximal torus of the unit group, and the Zariski closure of T in the whole matrix monoid Mn. In this paper we explicitly determine the idempotent lattice , the Renner monoid , and the cross section lattice Λ of MSOn in terms of the Weyl group and the concept of admissible sets (see Definition 3.1). It turns out that there is a one-to-one relationship between and the admissible subsets, and that is a submonoid of , the Renner monoid Mn. Also Λ is a sublattice of Λn, the cross section lattice of Mn.