C HAP T E R 2 ARITHMETICAL SEMIGROUPS AND THE GENERATING SETS OF PRIMES

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1 C HAP T E R 2 ARITHMETICAL SEMIGROUPS AND THE GENERATING SETS OF PRIMES 2.0 Introduction In this chapter we recall the definition of an arithmetical semigroup, generated by a set of 'primes', due to Knopfmacher [1 7 and show that the members of the same arithmetical semigroup can be generated by different sets of primes through an associated multiplication operation. For example, we can generate the arithmetical semigroup of rational positive integers by taking the products of povrers of the 'e-q.ary primes' that we introduce, with the exponents of these powers restricted modulo a rational positive prime q, (sections 2.2 to 2.5). We start with Knopfmacher's general setting of defining arithmetical semigroup and exhibit Z+ as an arithmetical semigroup in various ways through different product

2 21 operations, viz., those associated with the Narkiewicz type of set up L227 and e-q.ary set up, besides the one with usual product. 2.1 An Arithmetical Semigroup and its generating sets of primes We now give below the Knopfmacher's definition of an arithmetical semigroup and in the examples that follow exhibit how Z+ can be viewed as an arithmetical semigroup in various ways, by altering the associated product operation. Definition (Knopfmacher L1 7) Let G denote a commutative semigroup with identity element 1, relative to a multiplication'operation denoted by juxtaposition. Suppose that G has a finite or countably infinite subset p(g) (whose elements are called the primes of G), such that every element n f 1 in G has a unique factorisation of the form a n 1 = P1 a p r r where Pi's are distinct elements of p(g), the ai's are

3 22 positive integers, r may be arbitrary, and uniqueness is understood to be only upto the order of the factors indicated. Such a semigroup G will be called an arithmetical semigroup, if in addition there exists a realvalued norm mapping,;/ on G such that (2.1.3) (2.1.4) f 1 1 = 1, IpI> 1 for p E. P(G) 1mn I = [m 11 n' for all m, n G, and (2.1.5) the total numberng(x) of elements n e: G of norm Inl ~ x is finite, for each x> o. It can be verified that conditions (2.1.3), (2.1.4) and (2.1.5) are equivalent to the conditions (2.1.3) and (2.1.4) together with the condition that (2.1.5)' the total number 1!;(x) of elements pe:p(g) of norm Ipl~x is finite, for each x>o. In this let us call the subset p(g) of G as the set of generating primes for G. Each a j here in (2.1.~) runs

4 23 through all positive integers to generate G. Example. An example of this is obtained, when we take the set Z+ of all positive integers with P = R p ' the set of all positive rational primes, ~ssociated with the usual canonical representation of n and this arithmetical semigroup is denoted by G(Z+). We can also take G(Z*) = Z* (non-zero rational integers) likewise. In both of thesetafis'equal / to the usual modulus for a E Z-)E-. Example Narkiewicz type of arithmetical semigroup. A more general interesting situation arises by taking a Narkiewicz type of set-up ~57. Here we associate with each positive integer n, a set of its divisors A(n) (i~~ a subset of the set of all positive divisors of n) with a one-to-one correspondence between the numbe rs n E Z+ and the sets of divisors A(n) in a way that they satisfy the

5 24 following conditions: (i) (ii) (iii) (iv) d e A(n)~ ~ A(n) d E: A(m), m A(n)~ d ~ A(n), ~ c: A(~) t 1,n}~ A(n) If n = pk, p~' R p ' k E: Z+, then k ~ t 2t A(p) = 1 1, P, P,...., for some t Z+, with 0:; s ~ ~ ~ ~ and where s t s t P 1 A(p 2 ) for O~ s1 ~ s2~ s and such that (v) A(mn) = A(m) X A(n) whenever (m,n) = 1. Then we (vi) PA(Z+) = {pt 1 can see that with + t~z,p R p for which A(pt) = {1, pt }J \ we get an arithmetical semigroup G A (Z+) generated by PA with every positive integer n represented uniquely (upto order of factors) by

6 25 n::: t P a n(pt) t P A where the exponents of pt,s; ie, at's need not all run through the entire Z+; each may run through a corresponding associated subset It of Z+; ie, different aj's may run through different subsets; but each subset should contain zero. This has an implicit product operation associated with it. In l'uct, if pt P A, then there exists 8. set It of integers s E Z+ such that pt A(pst) and if r (~ro) is the greatest such s, then It ::: { 0, 1, 2,...., r } Note that the set P A together with the collection of the sets { It I pte PA' V p E: Rp } detcr~ines the representation of n in the canonical form related to this context, uniquely.

7 26 Here is an arithmetical semigroup of a general nature which arises in the set-up of Narkiewicz For each t for which pt PA' t:::. It = { t u I u : It} is an arithmetical progression and every pair of these progressions will have only i o}as their intersection. Note that as a set GACZ+) is the same as Z+; the way it is generated in GCZ+) is different from the way it is done in G A CZ+)., GCZ+) is generated by Rp, whereas GACZ+) is by PACz+) as generalised set of primes (or the set of primitive prime powers in the wording of Narkiewicz). The exponents in G(Z+) run through the entire Z+ for every p e:rp' whereas in GA(Z+) for each p~ PA we have an associated set It of exponents and for some (or all) t, It need not be entire Z+. ~\(A.I M~ ~'YY\~/v.~~~~, To take a particular example of this type, 1\ we may choose (p.87 Narkiewicz L257)

8 27 (2.1.8) P A = {p3, p4]u[. pt ltfz+,-{3' 6,9,12,4, SJ) with = { pt I tet, PERp) T = [ 3, 4} u{z+ - {3, I 3 = { 0, 1, 2, 3, 4] ) 6,9,12,4, s})) and I 4 = { 0, 1, 2 } ) It = { 0, 1 ] ) for tez+ - [3,6, 9, 12,4, 12 [ Here A(p )= 1, P, P, P, 8 ] 12 ] P Then each n E Z+ is uniquely expressible in the form n = If d E:A(n) we call d an A-divisor of n and write d fan. 2.2 e-q.ary type arithmetical semigroups Less complicated, easier and elegant examples are those in which, in the representation (2.1.2), we allow only a.'s J

9 to run through the non-negative residues to a rational prime modulus q and generate the elements of the arithmetical semig~oup denoted by G (Z+) wherein exponents are q restricted modulo q. Even then we can get a unique representation of n(upto order of factors) as product of powers of a suitable subset Pq(Z+) of Z+ = Gq(Z+) which we call the set of primes e-mod q (ie. exponentially q.ary generating set of primes) generating Z+, with exponents mbdulo q. We shall call, in short, P (Z+) as the set of primes e-mod q. q for Z+. Here Jalis usual modulus. We may omit the term e-mod q, whenever it is understood from the context. In fact, this can be done with any rational integral modulus q for exponents; not necessarily a prime modulus. We illustrate this in the cases q = 2, 3 and general q.

10 e-binary example (q = 2) a rational prime, i a non-negative rational integer]. which is a subset of Z+. n = IT IT' -p~i j i J Then if n E: Z+, ie-i(a}, jej={1, 2,..., r (arbitrary~, where I ). is a suitable set of non-negative rational integers'. In r a. fact, let n = rr p.j be the usual canonical represen. 1 J= J tation of n. We can express each a j in terms of its binary representation; ie. in the scale of 2. Thus a j =L: 2 i, where the summation is over for brevity, with ""i u. ' j = 1, 2,...? r J

11 30 Here u. will represent the number of places having 'one' J in the (unique) representation of a j in the scale of 2. Then obviously we have a unique representation (2.3.1). We call this unique representation of n as e-binary canonical factorisation of-no We may call Pb(Z+) as the set of e-binary (rational positive) primes. Thus tn' has a representation as a product of (powers of) e-binary primes relative to this context, (ie. with exponents modulo 2), which is unique upto the order of factors. Thus we have an arithmetical semigroup Z+= G 2 (Z+). Let C 1 j ' c 1 ), Z 2 1 ~ i E 1 j, - j I: 1, 2,. r. Then will be the binary expansion of some b j ~ a j and so d = IT b j. p. = IT J J j gives us a divisor of n, which we i 1." j J J call an e-binary

12 31 divisor of n. We note that when I j ' = for each j we get d ~1and 1 is an e-binary divisor of n. We denote the set of all e-binary divisors of n by D(n). Here a one-to-one onto correspondence between the sets BCn) and the positive integers n exists. We note that if each I.' is empty, then d = 1. Thus J B(n) ={d = IT rr j i 2 i Pj,where = f 1, 2,..., r}] If d E BCn) we write d {bn. 2.4 e-ternary example (q = 3) Let PtCZ+) = { p 3i l p a rational positive prime; i a non-negative rational integer) which is a subset of Z+. Then if n E Z+,

13 32 n = non-negative integers. To put it in detail, let us l2.' \.2.)~ express each a. in the 'scale of 3, so that J... ~ -1 a. = 1. ~ J i e 1 1 ) where o ~ i 11 "".... i (aj) = { i 21, i ,.,.~i2V j 1 1 ) n 1 2 ) #= 1 1 ) '# 1 2 ) = (the empty set), = u j ] J' = 1,2, r, = v j

14 33 We then obtain the unique representation in the form given in (2.4.1). We call this unique representation of n as e-ternary canonical factorisation of n. We may call Pt(Z+) as the set of e-ternary primes related to this context. Thus n has a representation as the product of powers of e-ternary primes, with exponents (r;lodulo 3) which is unique upto order of the factors. That exponents in this unique factorisation run modulo 3 is understood in the e-ternary product. We.thus have the arithmetical semigroup G 3 (Z+) = Z+. Let c 1 1 ' ) C 1 1 ); C 1 21 ' ) c 1 2 ); ( ) 1 22 ' ) C 1 2 ), where 1 21 ) n 1 22 ' ) =. Then some b. ~ J ~ a. will have the representation J

15 34 = 1.L: 3 i + ie 1 1 ' ) U 1 21 ' ) 2. L:.'3 i i~i22t(aj) and so any ternary divisor d is given by (2.4.5) d = = Uk note that 1 is an e-ternary divisor of n, obtainable empty set. We denote the set of all e-ternary divisors d of n by T(n). If de. T(n), we write d It n. (2.4.6) 't(n) := H(rr (1+ p3 i ) IT (1+ p3\ j~fi1(a) J i E I 2 ) J Members of T(n) are just the various terms in the expansion of Gt(n) and

16 T(n) 35 =fd n jd= GIT p~i j E. J i E 1 1 ' U 1 J e-q.ary example (q not necessarily a rational prime, but any positive integ~r) Let P q(z) ~ {pll p a rational prime, i a non-negative rational integer}, which is a subset of Z+. Then if n Z+, n = ]11 r we can wri'te, \::>O\.hu,iS~ d'sdtl w.t where It(a.) are suitable sets of non-negative rational J A '):.,... ",0 c- f- l'v\ t:l' ~. ~) we, integers. * ill;:') express a j in terms of its q.ary representation; ie. in the scale of (rational positive integer)q. Thus we a. = J obtain' 1L: qi i e 1 1 ) + (q-1)~ qi iei 1(a.).J q- J

17 36 where :. It ) ={i t1, i t2,..., i tutj } 0:= i t1 < i t2.... </!:. i tutj..' t = 1, 2, 3,...., (q-1 ) ; I (a.) n I (a.) =, x J y J x f: y, x, Y E: { 1, 2,..., q-1}; # It ) = "tltj' t = 1,2,,(q-1),j=1, 2,,r. We then obtain the unique representation of n in the form given in (2.5.1). We call this unique representation of n as e-q.ary canonical factorisation of n. We may call P (Z+) as the set of e-g.ary primes. Thus n q has a unique representation as a product of (powers of) e-q.ary primes (with exponents modulo q) upto order of the factors.

18 37 Remark In fact, if G is any general arithmetical semigroup such that peg) is the set of primes generating it, with positive integral exponents without any modular restriction for the exponent and q:any rational positive integer, we can get the corresponding set P (G) q of e-q.ary primes and if n G has the usual canonical representation given in (2.1.2), it will have the e-q.ary unique factorisation given by (2.5.1) except that here p E: peg) (instead of Similarly, for any arithmetical semigrc:up G, with primes :e and canonical representation (2.1.2) with aj's running through entire Z+, we can find, relative to any A-convolution 1227, a generalised set PA(G) of primes called

19 A-primes or A-primitive elements, to generate the same arithmetical semigroup G A with exponents at It, a suitable subset of Z+ as in (2.1.7).It is enough if we restrict to the casei of Z+ I with respect to various convolutions.