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By Brian G. Wybourne

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Let a E 4). Then (i) dim Ua = 1 and there is an isomorphism ea: (K,+) -, U. such that for all t E T, x e K, tea(x)t 1 = ea(a(t) x). (ii) For all a e W, alUaa = Uaa' (iii) G is generated by Ua(a E 4)) and T. 48. a e 4) is positive (relative to B) if Ua c B. Let + = (a E 4) a is Then 4 = - i+ is the set of positive roots relative to B positive relative to B. ) Moreover 4) is the disjoint union of 4+ and 4) Let A = A(B) = -d(B) _ (a E 4+ a is not a non-negative linear combination of 4)+\(a)). Then A is called the h= of 4), relative to B.

Hence Je A Jf = J. 11. The open problem then is to determine all possible 2l (S). If e is an idempotent in the maximum $-class of S, then 2l (S) __ 2e (M) where M = eSe. 1). When 2l (M')\(0) has a minimum element, the possibilities are determined in [89]. 1]. 12. Let S be a connected semigroup with zero 0. Then the following conditions are equivalent. (i) S is completely regular. (ii) S has no non-zero nilpotent elements. (iii) S is a monoid and the group of units of S is a torus. (iv) S is isomorphic to a closed submonoid of some t'In(K) with 0 being the zero matrix.

Generalize the Krohn-Rhodes decomposition theorem for finite semigroups [39] to linear algebraic semigroups. The more recent approach of Rhodes [103] might also be relevant The following result is well-known [16]. 15. Let M be a linear algebraic monoid. Then M is isomorphic to a closed submonoid of some ,AGn(K). Proof. We may assume that M is a closed subset of some Kd. ,fm from M into Kd, gl'"''gm E K[S] such that for all a,b E M, m ba = gi(a)fi(b) i=1 Let V denote the vector space of all maps from S into Kd.

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