The ring of regular functions of an algebraic monoid
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Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M=G_antM_aff, where M_aff is the affine submonoid M_aff=\bar{G_aff}. We then use this decomposition to calculate $\mathcal{O}(M)$ in terms of $\mathcal{O}(M_aff)$ and G_aff, G_ant\subset G. In particular, we determine when M is an anti-affine monoid, that is when $\mathcal{O}(M)=K$.