Abstract
In this paper we study connected regular linear algebraic monoids. If $\phi :{G_0} \to {\text {GL}}(n,K)$ is a representation of a reductive group ${G_0}$, then the Zariski closure of $K\phi ({G_0})$ in ${\mathcal {M}_n}(K)$ is a connected regular linear algebraic monoid with zero. In $\S 2$ we study abstract semigroup theoretic properties of a connected regular linear algebraic monoid with zero. We show that the principal right ideals form a relatively complemented lattice, that the idempotents satisfy a certain connectedness condition, and that these monoids are $V$-regular. In $\S 3$ we show that when the ideals are linearly ordered, the group of units is nearly simple of type ${A_l},{B_l},{C_l},{F_4}\;{\text {or}}\;{G_2}$. In $\S 4$, conjugacy classes are studied by first reducing the problem to nilpotent elements. It is shown that the number of conjugacy classes of minimal nilpotent elements is always finite.
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