Orders of the Renner monoids

In this paper, we give a general formula for a Renner monoid of a reductive monoid with zero. It is based on Putchaʹs type map. We apply this formula to find the orders of the Renner monoids for -irreducible monoids where G is a simple algebraic group over an algebraically closed field K, and is an irreducible representation associated with an arbitrary fundamental dominant weight.