POQORZELSKI in Providence, R.I. (U.S.A.) This paper deals with an initial development of a certain fragment of word arithmetic in a denumerable alphabet which is similar in many ways to the fragment of elementary number theory concerning the prime numbers. The development is based on cert2in generalizations of the primitive recursive word functions introduced by V . V U~K O V I~ [3]. The paper culminatzs with a theorem on the unique resolution of words of a word system in a denumerable alphabet, into word products of certain words, called primitive words, of a recursively definable subclass of the above-mentioned word system.We assume our previous note [Z] on the arithmetization of a word system in a denumerable alphabet as a prerequisite t o the present, paper; we shall refer to it as Note I. As in Note I, we shall be concerned with the denumerable alphabets A = { a l , as, . . .} and M = {m,, m2, . . . ), where M is the recursively definable class of MYCIELSKI numbers, their respective word systems Q (A) and Q (M) and respective empty words A and 1 . Again, the class of natural numbers is denoted by N . AS A I ( X , A ) = X , A',(X, a J ) = a f ( v , r )The author should like to acknowledge his thanks to V. V U~K O V I~ for his counsel.
This note presents a new arithmetization of a word system in a denumerable alphabet, based on a certain recursively definable class of natural numbers, introduced by JAN MYCIELSKI [l], which is an extension of the class of consecutive prime numbers.Throughout this note we employ the following notation: m n (a modified BOUR-BAKI notation for negation) ; A (conjunction) ; v (disjunction) ; *+ (equivalence);A (universal quantifier) ; V (existential quantifier) ; p (operation of minimalization). We shall also have occasion to use the following equality relation for sequences of natural numbers :E ( x , y; 1,1).+*x, = y1, E ( x , y; 1 , n) false for n > 1 , E ( x , y;m, 1 ) false for m > 1, E(x, y; m + 1 , n + 1) +=+ ( z m + i = Y~+ I ) A E ( x , y; m , n)
I(ΛΓ/, ΛΓy; 1, n) false for n> 1, \(xi, XJ; m, 1) false for m> i, \{x iy Xj; m + 1, n + l)OVv^n{x im + 1 = Xj v } A I(ΛΓ/, X" m, ra). §1. Word systems V(A), V(P). Firstly, we construct the formal commutative word system V( A ) introduced by Vuckovic.
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