HARJU and KARHUMÄKI

Proof . The first inequality is obvious. The second л -п" one follows from S g S Ç: S . Indeed, this implies

S S (S) S (S ) = s , and hence (S) = S = (S ),

We continue the analysis of (4) further. Denote by S^ the set of infinite words, w = s*.s^»... , made up from the elements of S. Define R^. Ç SxS by

( s , s» ) G R^ iff sS^^ n s'S^ ^ 0,

and let R^ be the transitive closure of R^. Again, R^ is a left congruence with finite index. From the tions we have ind(;,R^ ^ ind^^R and thus ind^^R^ < rank(S).

The next result is a generalization of a result of [12]: If an element of S^ has two different tions then deg(S) < rank(S).

THEOREM 8,

For each F-semigroup S, deg(S)< ind^^R <

£ rank(S),

Proof . Define a chain of F-semigroups in analogy to

( Д ) as follows: Sq = S, S^_^^ = [S^ U {s' : (s,ss') e Rg

for some s,ss* in S. ^ S. } ] . Again, rank(S.^) S

й rank(S.) for i ^ 0 and there is an index к so that ^ 1

S , = S, . for all i ^ 0. This S, is free and, moreover, R^ is the equality in S, , so that deg(S) й rank(S,) = indç. R . Now, the proof is complete if we show that indç R = indç R for i = 0. But this is due to the

l + l 1

fact that, with the notation above, if S. -, has a new generator then it must be s*, which replaces ss\ and s' cannot form a new equivalence class, since (s,ss*) € r¥ in S. implies that there exists s" € S. such that

0 '^ h.V

( s» , s " ) € RjiJ in S.

COROLLARY 4. Suppose for s,s* С S ^ S^ a_nd u € F we have s ;^ s * , uS^ П sS^^ ;^ 0, and uS^ П s'S^ ж 0. Then S i_s simplif iable .

Proof . If u € S, then Theorem 8 can be applied rectly. Otherwise consider S* = [S U {u}]. In this case we have indg,R^^ < rank(S') - 2 = rank(S) - 1 and the

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