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Consider an alphabet C2 = {aI,. . 6) is satisfied, and the maximum of the numbers I f ( a ) I , where a E C, and Ig(a)l, where a E C2, is less than or equal to P. Hence, the simplifiability of h can be decided by considering finitely many cases. ) 18 1 SINGLE HOMOMORPHISMS ITERATED It is also an immediate consequence of the definitions that a product of morphisms is simplifiable if one of the factors is simplifiable. The following necessary condition for a homomorphism to be elementary is sometimes very useful.

Aj:',Ay', (I) . . ,", . . -", . . , A 'IP, --] I ' } , = A p . . :', and h is defined by the productions I (11, = Alp1 = 1). . A'P1,-I z, z,= ' ' ' . A"', I1 A;," + A ) ? ' .. ' --t A\]', A']l' -+ A\2', I) + ' (I)p-;l(I)p-i2 ' .. 1, E ( G ) is docally catenative with cut p . 1) assures that / I is well defined, the theorem holds. 0 The locally catenative property is a global property of a DOL system in the sense that its formulation does not depend on the set of productions of the I 22 SINGLE HOMOMORPHISMS ITERATED system.

3 that there is no PDOL system generating the same word sequence or language as G. Let us consider the same problem from the point of view of length sequences. Clearly, the lengths of the words OJ, in E(G) satisfy I(I);I = 2It2 for all i 2 0. Thus, the very simple PDOL system G I = ( { a } , { U + a ' } , u4) generates the same word length sequence as G. , IQ; 1 < [ m i + I for all i ) such that there is no PDOL system G , with the same word length sequence as G ? The existence of such DOL systems will be shown in Chapter I l l .

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