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The rank is a decreasing function in the following sense: rank(xy) and rank(yx) Ä rank(x), for any x; y 2 Md . y/ D f0g. x/. 20. 20. k D 0; 1; : : : d / are two-sided ideals of Md . x/ D f0g. The idempotents of Md are by definition projection matrices e 2 D e. e/. 20(ii). The idempotent e is an orthogonal projection if and only if it is symmetric (see, for example [18]). Conversely, any projection operator on

13. Any finite 0-simple semigroup is completely 0-simple. Proof. We must establish that there is a nonzero idempotent because then there will be a minimal one too. Every element of S generates a subsemigroup with an idempotent. If 0 is the only idempotent, we can conclude that all elements of S are nilpotent: There is a k such that ak D 0 for all a 2 S . This implies that the finite semigroup S itself is nilpotent, since a … aS for all a 2 S : a D as implies a D as k for k D 1; 2; : : : and hence a D 0.

The minimal ideal in (ii) is f0g. S has divisors of zero, for example,  a a a a Ã à  à a a 0 0 D : a a 0 0 Idempotents of rank 1 are  and  à 1=2 1=2 1=2 1=2 1=2 1=2 à 1=2 : 1=2 Subsemigroups of matrices of rank Ä 1 is not 0-simple, since  1=2 1=2 à  à 1=2 1=2 1=2 …S : 1=2 1=2 1=2 (iii) S has of course a zero. < ; /. ) 38 1 Semigroups Idempotents are projections onto some subspace of

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50 Jahre Deutsche Statistische Gesellschaft Tagungen 1961 und 1962


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