Combinatorics on Words. Progress and Perspectives by Larry J. Cummings

By Larry J. Cummings

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Express v as a concatenation vrLvR. By the previous lemma Inf(ML,vL) is an essential class SL of ML and Inj{MR,vR) is an essential class SR of MR. SR. Now u is a random word so W W L R occurs as a block somewhere in u. Thus, there is a concatenation u=urLuR with wL a prefix of uL and wR a prefix of vR. ). Since u € ^ ^ , is in F and thus is in F. But v=vrLvR was an arbitrary concatenation so M accepts v. This shows that Αν\=φ. 5. order sentence φ: The following are equivalent for a monadic second 56 Kevin J.

Since , ML=, Mfr=f that accepts the ^-words in ^ ; in particular, M accepts u. Express v as a concatenation vrLvR. By the previous lemma Inf(ML,vL) is an essential class SL of ML and Inj{MR,vR) is an essential class SR of MR.

Section 1 contains preliminaries and a more detailed account of the ideas mentioned here. Section 2 contains a brief description of Fraisse-Ehrenfeucht games and their applications. Section 3 contains most of the main results. Section 4 contains the results on end model completeness. I would like to thank Jan Mycielski for many helpful suggestions which have improved the exposition in this paper. ON RICH WORDS 41 1. PRELIMINARIES Fix a finite alphabet Σ. , a pair , where A is a set and < is a total order on A.

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