New PDF release: Varieties of Formal Languages

By J. E. Pin (auth.)

ISBN-10: 1461293006

ISBN-13: 9781461293002

ISBN-10: 1461322154

ISBN-13: 9781461322153

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The following conditions are equivalent. (i) (ii) (iii) (iv) (v) (vi) D is regular. D contains a regular element. Each Bl-class of D contains at least one idempotent. Each If-class of D contains at least one idempotent. D contains at least one idempotent. There exist x, y E D such that xy E D. Proof If a = asa, then a fJl e where e = e2 = as. Conversely if aBle, where e is idempotent, there exists u E Sl such that au = e and therefore a = ea = e2 a = auea. Likewise a = asa if and only if La contains an idempotent.

L' e if and only if ae = a. ;, fax then a,l ax and a &l ax. ;, f xa then a,l xa and a 2 xa. ;, fJi axy then a, ax and axy are &l-equivalents. l' yxa then a, xa and yxa are 2 -equivalents. l' b and if a,l b then a 2 b... If a ::';'fJib and if a,l b then afJl b. Proof (1) If a ::';'fJie, there exists UES 1 such that a = eu. Hence we can deduce ea = eeu = eu = a. l' e is similar. ;, f a, we have in fact a,l ax and there exist u, v E Sl such that uaxv = a, whence uka(xv)k = a for every k> O. Let us choose k such that e = Uk is idempotent.

Proof We shall show that, if L= XA* u Ywith X and Yfinite, then S(L)EK. Since S(Y) is nilpotent and therefore an element of K, it is sufficient to establish that S = S(X A *) is in K. Let n be the maximum length of words of X and let u be a word of length greater than or equal to n. Then for every word vEA*, we have uv '" XA* U. It is clear in fact that xuvYEXA* ~xuEXA* ~xuYEXA*. Hence we deduce that ts = t for every t E sn and for every s E S, and therefore in particular es = e for every e E E(S).

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Varieties of Formal Languages by J. E. Pin (auth.)

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