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Let Q be a Schreier system for H and X={paq\p9qeQ,aeA9paG(H-l)q}\ Show that X generates H. Problems 15 c. Show that each paqE. X is reduced as written and that, in the product of two elements of XUX the letters a in the triple (p,a,q) never cancel unless the whole product does. d. 2). 2. A submonoid N of A* is generated by a prefix iff it satisfies: for all m,nE A*. Such a submonoid is called (right) unitary. 3. Let P be the set of words P = {WW\WEL A*} . , u = u) of even length. Show that the submonoid P* is right and left unitary.
This is the motivation for introducing the concept of a formal series. Let A' be a ring with unit; in the sequel K will be generally be the ring Z of all integers. A formal series (or series) with coefficients in K and variables in A is just a mapping of the free monoid A* into K. The set of these series is denoted by K((A)). For a series o€zK((A)) and a word wGA*, the value of a on w is denoted by (a, w) and called the coefficient of w in a; it is an element of K. For a set X C A*, we denote by X the characteristic series of X, defined by The operations of sum and product of two series defined by: (a + T, H>> = (a, w) + (T, W) , (OT,W) = 2 O9TEK((A)) are
2(i)) = toJ+l'-vn. We may assume s ^ l , since otherwise j -1 ^ 1 and we can replace Vj by t^-j. Next, define^' and z' by As mentioned before, ^' and z' are nonempty. 2) Now, we derive a contradiction by showing that w contains a square. Consider first the case where yt = \. In this case, t>, = y', t>• = s, whence by Eqs. 2) u = © , t y • • VJ_XVJ = »,+,••• ©„_,*'. 2. Occurrence of uu in «(w): (i) localization of ww, (ii) double factorization of w. 3 Since a(A) is a prefix code, this implies vx = t>y+,,..