Abstract
Let MSOn (n is even) be the special orthogonal algebraic monoid, T a maximal torus of the unit group, and T the Zariski closure of T in the whole matrix monoid Mn. In this paper we explicitly determine the idempotent lattice E(T), the Renner monoid R, 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 E(T) and the admissible subsets, and that R is a submonoid of Rn, the Renner monoid Mn. Also Λ is a sublattice of Λn, the cross section lattice of Mn.
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