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By W. Gunion Rutherford
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Example text
The “local variant” of this fact is not true, as it is shown by Theorem 13. Theorem 13. There exist a language such that x is simple, but not adequate. {r,P, @} and a word x E r PROOF. Let us consider the language used in the proof of Proposition 1 . As shown in this proof, no word of this language is adequate. On the other hand, we have S ( a ) = {a, c}, P(a) = { a } , P(c) = { c , d}. Hence P(c) fl S(a)= {c} # 0 and P(a) r l S ( a ) = { a } # 0. Therefore, a is a simple word. It is easy to see that there exist simple languages which are neither completely homogeneous nor perfect, whereas there exist completely homogeneous and perfect languages which are not simple.
Thus, by definition, any simple language is homogeneous. The converse is not true, as is shown by Proposition 16. Proposition 16. There exists a homogeneous language which is not simple. PROOF. Let {r,P , (a} be a language such that S(x) C P(x) for each r, whereas S is not the unit partition of r. (An example of such a language is {r,S, (a}, where r = { a , b } and (a = { a , b } . ) This language is completely homogeneous and, in view of Proposition 9, it is homogeneous. On the other hand, P(x) n S(x) = S(x) and, since S # E , there exists a word x1 such that S(x,) # { x l } .
Izd. Akad. Nauk SSSR, Moskow, 1962. 32. I. I . ” Izd. Akad. Nauk SSSR, 1963, pp. 94-1 03. 33. 1. I . Revzin, Some problems concerning the theory of language models (in Russian). Nauchn-tekhn. , N o . 8,42-46 (1964). 34. A. Sestier, Contribution B une thkorie ensembliste des classifications linguistiques. Premier congrts de 1’Association franCaise de calcul, Grenoble, 1960; Paris, I96 1 , pp. 293-305. 35. V. A. Uspenskii, On defining the part of speech in the set-theoretic system of the language (in Russian).