The ring of regular functions of an algebraic monoid

TL;DR

This paper extends the Rosenlicht decomposition from the unit group of an algebraic monoid to the monoid itself, enabling explicit calculation of the monoid's regular functions and characterizing anti-affine monoids.

Contribution

It generalizes the Rosenlicht decomposition to algebraic monoids and provides a method to compute their regular functions, also characterizing anti-affine monoids.

Findings

Decomposition M=G_ant M_aff extends Rosenlicht's G=G_ant G_aff.

Explicit calculation of (M) in terms of (M_aff) and G_aff.

Criteria for M to be an anti-affine monoid ((M)=K).

Abstract

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 $\mathcal{O}(M)=K$.