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Then N is an inverse semigroup. Certain finite inverse semigroups will play the rˆ ole of the Weyl group in the theory of algebraic monoids. 8. 1 Abstract Semigroups 1. 69 in fact yield completely (0-)simple semigroups. 2. Let S = Γ × G × Λ be a completeley simple semigroup with sandwich matrix P : Λ × Γ → G. Identify the Green’s relations R, L and H on S in terms of P . 3. Prove that Tn (K) is a semilattice of archimedean semigroups. 4. Prove that Mn (K) is sπr. 5. Prove that S = {x ∈ Mn (K) | rank(x) ≤ 1} is a completely 0-simple semigroup.

For e), let H denote the H-class of a. From a), we see that aHa2 . Then a2 x = a for some x ∈ S 1 . Then ai+1 xi = a for all i > 0. Thus ai Ra for all i > 0. By a) again, ai ∈ H for all i > 0. But there exist j > 0 and e ∈ E(S) such that aj He. But then e ∈ H and so H is a group. For f), suppose that aJabJb. Then by a), aRabLb. Hence there exist x, y ∈ S 1 such that abx = a and yab = b. Then ya = yabx = bx. Hence aya = a and bxb = b. Thus ya ∈ E(S) and aLya = bxRb. Conversely, assume that there exists e ∈ E(S) such that aLeRb.

30 of [82]. Thus gGe g −1 = hGe h−1 and so gGe Ru (G)g −1 = Ge Ru (G). Hence gN g −1 = N . 3 N is regular. Now Ge × Ru (G) −→ Ge Ru (G) is bijective, and its kernel is Ge ∩ Ru (G), which is an infinitesimal unipotent group scheme. 1 of [117], that Ge ∩ Ru (G) is actually central in Ge , yet Z(Ge ) is a diagonalizable group scheme. 12. For b) one checks that ϕ is surjective and birational while M is normal. Thus, ϕ is an isomorphism. 2 above that any normal, reductive monoid M is determined by the diagram T ⊇T ⊆G.

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