A reductive monoid M is called J-irreducible if M{0} has exactly one minimal G x G-orbit. There is a canonical cellular decomposition for such monoids. These cells are defined in terms of idempotents, B x B-orbits, and other natural monoid notions. But they can also be obtained by the method of one-parameter subgroups. This decomposition leads to a number of important combinatorial and topologial properties of the monoid of B x B-orbits of M. In case M{0} is rationally smooth these cells are closely related to affine spaces. They can be used to calculate the Betti numbers of a certain projective variety.
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