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We give a classification of noncommutative algebraic monoid structures on normal affine varieties such that the group of invertible elements of the monoid is connected, solvable, and has a one-dimensional unipotent radical. We describe the set of idempotents and the center of such a monoid and give a criterion for existence of the zero element. [ABSTRACT FROM AUTHOR]

Consider an algebraic semigroup S and its closed subscheme of idempotents, E( S). When S is commutative, we show that E( S) is finite and reduced; if in addition S is irreducible, then E( S) is contained in a smallest closed irreducible subsemigroup of S, and this subsemigroup is an affine toric variety. It follows that E( S) (viewed as a partially ordered set) is the set of faces of a rational polyhedral convex cone. On the other hand, when S is an irreducible algebraic monoid, we show that E( S) is smooth, and its connected components are conjugacy classes of the unit group. [ABSTRACT FROM AUTHOR]