The Ring of Regular Functions of an Algebraic Monoid Lex Renner and Alvaro Rittatore

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G = GantGaff ∼= (Gant × Gaff)/Gaff ∩ Gant, where Gant is the maximal anti-affine subgroup of G, and Gaff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M = GantMaff ∼= Gant∗Gaff∩GantMaff , where Maff is the affine submonoid Maff = Gaff . We then use this decomposition to calculate O(M) in terms of O(Maff) and Gaff , Gant ⊂ G. In particular, we determine when M is an anti-affine monoid, that is O(M) = k.