Orbit Structure Of The Symplectic Monoids Over An Arbitrary Field

Doty has defined the symplectic monoids SpM_n(K) over an arbitrary field and studied their G x G orbits stucture and discussed the irreducibility of the the symplectic monoids SpM_n(K) as algebraic varieties. In this paper, we will go further into the properties of the symplectic monoids and we will define the Weyl group and the Borel subgroup of a generalized symplectic monoid and the Renner monoid of a symplectic monoid. We will also discuss the cross-section lattice of the symplectic monoids and their Bruhat Decomposition.