OKNINSKI

migroup S the following assertion is true :

( x ) if K£sJ i£ regular and all principal factors of S are finite^ then S is^ finite. Then^ for any finitely generated semigroup S the rity of the algebra K[sJ implies that S i^ finite.

Proof . If S is finitely generated, then for some n>l

n

and elements a. ,ap,...a £S we have S = V^ J . Take 1 ^ n ^^^ a^

the shortest presentation of S of this form. Then, for

^ " J^vy uJ ^I the semigroup S/l "^ S is com- ^1 ••• ^n-1 ^n ^n

pletely 0 - simple since S must be regular,[19], and riodic, [I3j. Moreover, while all subgroups of S are cally finite, [19j, and S/I is finitely generated, then the Rees theorem implies that S/l is finite. Let M be the intersection of all ideals T of S such that S/T is finite. We will show that all principal factors of S/M are finite. If a6S\M, then there exists an ideal T of S such that a^T and S\T is finite. Let a^^, a^ note the images of a under the natural homomorphisms S—9S/M, S—>S/T respectively. Since the principal factor of a^, in S/M is isomorphic to the principal factor of am in S/T, then it must be finite. Hence S/M satisfies the hypotheses of (x) and so it must be finie. Suppose that Ж i ^, Prom ["13] > Lemma 3> it follows that M is a finitely generated semigroup. Thus, as above, we may find an ideal J of M with M/J finite. Prom [3]^ Theorem 2.41, it follows that J is an ideal of S. Plainly, s/j is finite which contradicts the choice of M. This means that M is empty and so S is finite.

REFERENCES

1 . Bourbaki N., Theories spectrales, Hermann, Paris, 19б7.

2 . Chekanu G.P., Semisimple locally finite algebras, dies in the theory of rings, algebras and modules. Mat. Issled. No. 76(1984),172-179.

3 . Clifford A.H. and G.B.Preston, The algebraic theory of semigroups, vol.1. Providence^ 196I.

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