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By Samuel Eilenberg

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Extra resources for Automata, Languages and Machines. Volume B

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Define Li = {S I s E S, QS c A i } for 1 5 i 4 k . Then Li is a left ideal in S and the maximality of the sets Ai implies s = L , u ... ULk Define T = L, L = L,, U ... ULk Then T is a subsemigroup of S and L u T = S . 5 yields X < Z o P with Z=(Q,L), If A, = Y=(Tul,,T) L , . Thus card L < card S and Qs, then s 217 If A, = Qt, then t Consequently $ 1. Thus t y17 @ T and card T < card S.

S’ = {s‘ I s‘ E PF(Q), s’ c s for some s E S } Show that X - X . 3. Show that X mp Y 12X = S‘. Y and both 9 and 9-l wP, are proper. 8. Join, Sum, and Direct Product Let Q , , Q2 be finite sets and let be their disjoint union. T h e semigroups PF(Q,) and PF(Q,) will both be regarded as subsemigroups of PF(Q), PF(Q,) consisting of all partial functions s: Q -+ Q such that Q1s c Q1, Q2s = 0 . Similarly for PF(Q,). We define the join of X I and X, to be with Q = Q1 u Q, ind S the subsemigroup of PF(Q) generated by Sl U S 2 .

Show that if X c Y and Y c X are p r o p inclusions, then X = Y. For any inclusion X c Y , show that X c Y I Qx. the inclusion X c Y is proper, then also X c Y,. 8. 9. Show that X I 0 = 0 if Sx = 0 and X I 0 = 0' other- wise. 10. Let T be a monoid in the semigroup S. Show that T = ( T , T ) c (S, T ) c (S, S), What can be said c (S, S)I T if T is onIy assumed to be a subsemigroup of S? 7. Isomorphisms and Equivalences Let X and Y be ts's and let y : Q y w Qx be a bijection. We shall say that q is an equivalence (notation: Y wVX ) if X

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