# Download Automata, Languages and Machines. Volume B by Samuel Eilenberg PDF

By Samuel Eilenberg

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X X. Proof. QX Let Q = { q l , . . ,q k } and Qk be the k-fold Cartesian product Define the function . . x Q. i: S + Q k si = (qlS, li = ( q l , * * * 9 qks) . . , qk) if s' # S Clearly i is injective. For each s E S , t = (s, . . ,s) is a transformation of X ' k )and xit = mi. 9 If X = (Q, S ) is a monogenic ts, then x~~
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~~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 . Explicitly we have S= S, Sl s 2 u S, u 8 if if if S, # 0 # S, S, = 0 S, = 0 8. Join, Sum, and Direct Product 19 The following relations are clear Xi c X,vX, for i = 1 , 2 x,vx,=x,vx, ( X , v X,) v x, = x, v ( X , v X 3 ) xvo=x X, < Y, and X , < Y, imply X,vX, < Y,v Y, xl*v x,. ~~

If X and Y are transformation groups then so is 0 Y. Proof. Let (f,t ) be a transformation of Define g : P S by setting XoY with f : P -+ S. ---f Pg = W - ' l f I-' Then k,P N f , t)(g,t-l) = ( d P f 1, pt)(g, t-l) = M P f )(Pf )-l, ptt-') = (g,P) Thus (g, t-l) is the inverse of ( f , t ) In Section 7 we denoted by TS the set of all equivalence classes of ts's. TS had then the structure of a (partially) ordered set. A glance at the list of formal rules shows that using the wreath product X o Y as a multiplication, TS is a monoid with the class of 1' as unit element.