A ‘linear algebraic monoid’ is an affine variety defined over an algebraically closed field \(K\) with an associative morphism and an identity. The unit group of an algebraic monoid is an algebraic group. An algebraic monoid is ‘irreducible’ if it is irreducible as a variety. An irreducible monoid is ‘reductive’ if its unit group is a reductive group. Let \(M\) be a reductive monoid with unit group \(G\) and a zero element \(0\). Let \(B\subset G\) be a Borel subgroup, \(T\subset B\) be a maximal torus, and \(W=N_G(T)/T\) be the Weil group. Let \(\overline{N_G(T)}\) be the Zariski closure of \(N_G(T)\) in \(M\). Then \(R=\overline{N_G(T)}/T\) is an inverse monoid with unit group \(W\). The monoid \(R\) is called the ‘Renner monoid’.
There is given a general formula for the order of a Renner monoid of a reductive monoid with zero. It is based on Putcha’s type map. This formula is applied to finding the orders of the Renner monoids for \(\mathcal J\)-irreducible monoids \(\overline{K^*\rho(G)}\) where \(G\) is a simple algebraic group over an algebraically closed field \(K\), and \(\rho\colon G\to\text{GL}(V)\) is an irreducible representation associated with an arbitrary fundamental dominant weight.