The [formula omitted]-class structure of connected algebraic monoids

Abstract

Let S be a connected algebraic monoid and let U ( S) denote the finite lattice of all regular I -classes of S. The main theorem of this paper is that U ( S) is relatively complemented if and only if S is a semilattice of archimedean semigroups. If further S has a zero, then U ( S) is relatively complemented if and only if the group of units G of S is solvable. Another characterization for the solvability of G is that the product of any two nilpotent elements of S is again nilpotent. The width and depth of any J ϵ U( S) are defined. Also, the width, depth and height of S are defined. These structure numbers are studied in some detail and the solvability of G is characterized in terms of them.