Linear size test sets for certain commutative languages

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We prove that for each positive integer n, the finite commutative language En = c (a 1 a 2 ...an ) 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) alph(w ) = alph}(L ); and (ii) each symbol a ∈ 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 = Card(alph(L )). The considerations rest on the analysis of some basic types of word equations.