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In this article, we study the Bruhat-Chevalley-Renner order on the complexsymplectic monoid $MSp_n$. After showing that this order is completelydetermined by the Bruhat-Chevalley-Renner order on the linear algebraic monoidof $n\times n$ matrices $M_n$, we focus on the Borel submonoid of $MSp_n$. Byusing this submonoid, we introduce a new set of type B set partitions. Wedetermine their count by using the ``folding'' and ``unfolding'' operators thatwe introduce. We show that the Borel submonoid of a rationally smooth reductivemonoid with zero is rationally smooth. Finally, we analyze the nilpotentsubsemigroups of the Borel semigroups of $M_n$ and $MSp_n$. We show that,contrary to the case of $MSp_n$, the nilpotent subsemigroup of the Borelsubmonoid of $M_n$ is irreducible.

On the Borel Submonoid of a Symplectic Monoid