## Linear size test sets for certain commutative languages.(English)Zbl 1010.68103

Summary: We prove that for each positive integer $$n$$, the finite commutative language $$E_n= c(a_1a_2 \cdots a_n)$$ possesses a test set of size at most $$5n$$. Moreover, it is shown that each test set for $$E_n$$ has at least $$n-1$$ elements. The result is then generalized to commutative languages $$L$$ containing a word $$w$$ such that (i) $$\text{alph} (w)-\text{alph}(L)$$; and (ii) each symbol $$a\in\text{alph}(L)$$ occurs at least twice in $$w$$ if it occurs at least twice in some word of $$L$$: each such $$L$$ possesses a test set of size $$11n$$, where $$n= \text{Card (alph}(L))$$. The considerations rest on the analysis of some basic types of word equations.

### MSC:

 68R15 Combinatorics on words

### Keywords:

finite commutative language; word equations
Full Text:

### References:

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