In this paper, we present a characterization of regular convolution

Sagi S.

Algebra and Discrete Mathematics

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

“adm-n1” — 2018/4/2 — 12:46 — page 147 — #149Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 25 (2018). Number 1, pp. 147–156c© Journal “Algebra and Discrete Mathematics”Characterization of regular convolutionsSankar SagiCommunicated by V. V. KirichenkoAbstract. A convolution is a mapping C of the set Z+ ofpositive integers into the set P(Z+) of all subsets of Z+ such that,for any n ∈ Z+, each member of C(n) is a divisor of n. If D(n) isthe set of all divisors of n, for any n, then D is called the Dirichlet’sconvolution [2]. If U(n) is the set of all Unitary(square free) divisorsof n, for any n, then U is called unitary(square free) convolution.Corresponding to any general convolution C, we can define a binaryrelation 6C on Z+ by ‘m 6C n if and only if m ∈ C(n)’. In thispaper, we present a characterization of regular convolution.IntroductionA convolution is a mapping C of the set Z+ of positive integers intothe set P(Z+) of subsets of Z+ such that, for any n ∈ Z+, C(n) is anonempty set of divisors of n. If C(n) is the set of all divisors of n, foreach n ∈ Z+, then C is the classical Dirichlet convolution [2]. IfC(n) = {d / d|n and (d,nd) = 1},then C is the Unitary convolution [1]. As another example ifC(n) = {d / d|n and mk does not divide d for any m ∈ Z+},then C is the k-free convolution. Corresponding to any convolution C, wecan define a binary relation 6C in a natural way bym 6C n if and only if m ∈ C(n).2010 MSC: 06B10, 11A99.Key words and phrases: semilattice, lattice, convolution, multiplicative, co-maximal, prime filter, cover, regular convolution.“adm-n1” — 2018/4/2 — 12:46 — page 148 — #150148 Characterization of regular convolutions6C is a partial order on Z+ and is called partial order induced by theconvolution C [11], [12]. W. Narkiewicz [2] first proposed the concept ofa regular convolution, and in this paper we present a lattice theoreticcharacterization of regular convolution and prove that the Dirichlet’s con-volution is the unique regular convolution that induces a lattice structureon (Z+,6C).1. PreliminariesLet us recall that a partial order on a non-empty set X is defined as a bi-nary relation 6 on X which is reflexive (a 6 a), transitive (a 6 b, b 6 c =⇒a 6 c) and antisymmetric (a 6 b, b 6 a =⇒ a = b) and that a pair (X,6)is called a partially ordered set (poset) if X is a non-empty set and 6 is apartial order on X.For any A ⊆ X and x ∈ X, x is called a lower(upper) bound of A ifx 6 a (respectively a 6 x) for all a ∈ A. We have the usual notations ofthe greatest lower bound (glb) and least upper bound (lub) of A in X.If A is a finite subset {a1, a2, · · · , an}, the glb of A (lub of A) is denotedby a1 ∧ a2 ∧ · · · ∧ an orn∧i=1ai (respectively by a1 ∨ a2 ∨ · · · ∨ an orn∨i=1ai).A partially ordered set (X,6) is called a meet semi lattice if a ∧ b(=glb{a, b}) exists for all a and b ∈ X. (X,6) is called a join semi latticeif a ∨ b (=lub{a, b}) exists for all a and b ∈ X. A poset (X,6) is calleda lattice if it is both a meet and join semi lattice. Equivalently, latticecan also be defined as an algebraic system (X,∧,∨), where ∧ and ∨ arebinary operations which are associative, commutative and idempotent andsatisfying the absorption laws, namely a ∧ (a ∨ b) = a = a ∨ (a ∧ b) forall a, b ∈ X; in this case the partial order 6 on X is such that a ∧ b anda ∨ b are respectively the glb and lub of {a, b}. The algebraic operations∧ and ∨ and the partial order 6 are related bya = a ∧ b ⇐⇒ a 6 b ⇐⇒ a ∨ b = b.Throughout the paper Z+, N , and P denote the set of positive integers,the set of non-negative integers, and set of prime numbers respectively.Theorem 1 ([12]). Let 6C be the binary relation induced by convolution C.Then(1) 6C is reflexive if and only if n ∈ C(n).(2) 6C is transitive if and only, for any n ∈ Z+,⋃m∈C(n) C(m) ⊆ C(n).(3) 6C is always antisymmetric.“adm-n1” — 2018/4/2 — 12:46 — page 149 — #151S. Sagi 149Corollary 1 ([12]). The binary relation 6C induced by convolution Con Z+ is a partial order if and only if n ∈ C(n) and⋃m∈C(n) C(m) ⊆ C(n)for all n ∈ Z+.Definition 1 ([12]). Let X and Y be non-empty sets and R and S bebinary relations on X and Y respectively. A bijection f : X → Y is saidto be a relation isomorphism of (X,R) into (Y, S) if, for any elements aand b in X,aRb in X if and only if f(a)Sf(b) in Y.Theorem 2 ([12]). Let θ : Z+ →∑P N be the bijection defined byθ(n)(p) = the largest a in N such that pa divides n,Then a convolution C is multiplicative if and only if θ is a relation isomor-phism of (Z+,6C) onto (∑P N,6C).Theorem 3 ([9], [10]). For any multiplicative convolution C, (Z+,6C) isa lattice if and only if (N,6pC) is a lattice for each prime p.Now we state the following theorems on co-maximality and primefilters.Theorem 4 ([5]). Let (S,∧) be any meet semi lattice with smallest element0 satisfying the descending chain condition. Also, suppose that every properfilter of S is prime. Then the following are equivalent to each other.(1) For any x and y ∈ S, x‖y =⇒ x ∧ y =0.(2) S − {0} is a disjoint union of maximal chains.(3) Any two incomparable filters of S are co-maximal.Theorem 5 ([5]). Let C be any multiplicative convolution such that(Z+,6C) is a meet semi lattice. Then any two incomparable prime filtersof (Z+,6C) are co-maximal if and only if any two incomparable primefilters of (N,6pC) are co-maximal, for each p ∈ P .Theorem 6 ([3]). Let p be a prime number. Then every proper filter in(N,6pC) is prime if and only if [pa) is a prime filter in (Z+,6C) for alln > 0.Theorem 7 ([3]). A filter F of (Z+,6C) is prime if and only if thereexists unique p ∈ P such that F p is a prime filter of (N,6pC) and F q = Nfor all q 6= p in P and, in this case,F = {n ∈ Z+ | θ(n)(p) ∈ F p}.“adm-n1” — 2018/4/2 — 12:46 — page 150 — #152150 Characterization of regular convolutionsTheorem 8 ([3]). Let F be a filter of (Z+,6C). Then F = [pa) forsome prime number p and a positive integer a which is join-irreducible in(Z+,6C).Definition 2. Any complex valued function defined on the set Z+ of pos-itive integers is called an arithmetical function. The set of all arithmeticalfunctions is denoted by A.The following is a routine verification using the properties of additionand multiplication of complex numbers.Theorem 9. For any arithmetical functions f and g, define(f + g)(n) = f(n) + g(n) and (f · g)(n) = f(n)g(n)for any n ∈ Z+.Then + and · are binary operations on the set A of arithmeticalfunctions and (A,+, ·) is a commutative ring with unity in which theconstant map 0̄ and 1̄ are the zero element and unity element respectively.Definition 3. Let C be a convolution and f and g arithmetical functionsand C be the field of complex numbers. Define fCg : Z+ → C by(fCg)(n) =∑d∈C(n)f(d)g(nd).We can consider C as a binary operation, as defined above, on theset A of arithmetical functions. W.Narkiewicz proposed the followingdefinition.Definition 4 ([2]). A convolution C is called regular if the following aresatisfied.(1) (A,+, C) is a commutative ring with unity, where + is the point-wiseaddition. This ring will be denoted by AC .(2) If f and g are multiplicative arithmetical functions, then so is theproduct fCg (f is said to be multiplicative if f(mn) = f(m)f(n).)(3) The constant function 1̄, defined by 1̄(n) = 1 for all n ∈ Z+, is aunit in the ring AC .It can be easily verified that the arithmetical function e, defined bye(n) ={1 if n = 1,0 if n > 1is the unity (the identity element with respect to the binary operation C).“adm-n1” — 2018/4/2 — 12:46 — page 151 — #153S. Sagi 151W. Narkiewicz proved the following two theorems.Theorem 10 ([2]). A convolution C is regular if and only if the followingconditions are satisfied for any m,n and d ∈ Z+.(1) C is multiplicative convolution; i.e., (m,n)=1 ⇒ C(mn)=C(m)C(n).(2) d ∈ C(m) and m ∈ C(n) ⇔ d ∈ C(n) and md∈ C(nd).(3) d ∈ C(n) ⇒ nd∈ C(n).(4) 1 ∈ C(n) and n ∈ C(n).(5) For any prime number p and any a∈Z+, C(pa) = {1, pt, p2t, · · · , prt},rt = a for some positive integer t and pt ∈ C(p2t), p2t ∈ C(p3t),. . . ,p(r−1)t ∈ C(pa).Theorem 11 ([2]). Let K be the class of all decompositions of the set ofnon-negative integers into arithmetic progressions (finite or infinite) eachcontaining 0 and no two progressions belonging to same decompositionhave a positive integer in common. Let us associate with each p ∈ P ,a member πp of K. For any n = pa11 pa22 · · · parr , where p1, p2, · · · , pr aredistinct primes and a1, a2, · · · , ar ∈ N , defineC(n) = {pb11 pb22 · · · pbrr | bi 6 ai, and bi and ai belong to the sameprogression in πpi .}Then C is a regular convolution and, conversely every regular convolutioncan be obtained in this way.From the above theorems, it is clear that any regular convolution Cis uniquely determined by a sequence {πp}p∈P of decompositions of Ninto arithmetical progressions (finite or infinite) and we denote this byexpression C ∼ {πp}p∈P .Definition 5. For any two elements a and b in a partially ordered set(X,6), a is said to be covered by b (b is a cover of a) if a < b and there isno c ∈ X such that a < c < b. This is denoted by a− < b.We note that θ : Z+ →∑P N defined byθ(a)(p) = the largest n in N such that pn divides a,for any a ∈ Z+ and p ∈ Pis a bijection.2. Main resultsIn the following two theorems, we prove that any regular convolution Cgives a meet semi lattice structure on (Z+,6C) and the convolution C is“adm-n1” — 2018/4/2 — 12:46 — page 152 — #154152 Characterization of regular convolutionscompletely characterized by certain lattice theoretic properties of (Z+,6C).In particular Dirichlet’s convolution is the only regular convolution C whichgives a lattice structure on (Z+,6C).Theorem 12. Let C be a convolution and 6C the relation on Z+ inducedby C. Then C is a regular convolution if and only if the following propertiesare satisfied.(1) θ : (Z+,6C) →∑p∈P (N,6pC) is a relation isomorphism.(2) (Z+,6C) is a meet semi lattice.(3) Any two incomparable prime filters of (Z+,6C) are co-maximal.(4) F is a prime filter of (Z+,6C) if and only if F = [pa) for somep ∈ P and a ∈ Z+.(5) For any m and n ∈ Z+, m− <C n =⇒ 1− <Cnm6C n.Proof. Suppose that C is a regular convolution. By Theorem 11,C ∼ {πp}p∈P , where each πp is a decomposition of N into arithmeticprogressions (finite or infinite) in which each progression contains 0 andno positive integer belongs to two distinct progressions. For any a, b ∈ Nand p ∈ P , let us write for convenience,〈a < b〉 ∈ πp ⇐⇒ a and b belong to the same progression of πp.Since C is regular, C satisfies properties (1)–(5) of Theorem 10. From(2) and (4) of Theorem 10 and Corollary 1, it follows that 6C is a partialorder on Z+. Since C is multiplicative, it follows from Theorem 2 thatθ : Z+ →∑P N is an order isomorphism. Therefore the property (1) issatisfied. For simplicity and convenience, we shall write n̄ for θ(n). Foreach n ∈ Z+, n̄ is the element in the direct sum∑P N defined byn̄(p) = the largest a in N such that pa divides n.n 7→ n̄ is an order isomorphism of (Z+,6C) onto∑p∈P (N,6pC), where foreach p ∈ P , 6pCis the partial order on N defined bya 6pCb if and only if pa ∈ C(pb).For any m and n ∈ Z+, let m ∧ n be the element in Z+ defined bym ∧ n(p) ={0 if 〈m(p), n(p)〉 /∈ πp,min{m(p), n(p)} otherwise.for all p ∈ P . If 〈m(p), n(p)〉 ∈ πp, thenm(p) 6pCn(p) or n(p) 6pCm(p)“adm-n1” — 2018/4/2 — 12:46 — page 153 — #155S. Sagi 153and hence m ∧ n(p) 6 m(p) and n(p) for all p ∈ P . Therefore m ∧ n is alower bound of m and n in (Z+,6C). Let k be any other lower bound ofm and n. For any p ∈ P , if 〈m(p), n(p)〉 ∈ πp, then, sincek(p) 6pCm(p) and k(p) 6pCn(p),we havek(p) 6pCm ∧ n(p).If 〈m(p), n(p)〉 /∈ πp, thenk(p) = 0 = m ∧ n(p).Thus k 6 m∧n. Therefore, m∧n is the greatest lower bound of m and n in(Z+,6C). Thus (Z+,6C) is a meet semi lattice and hence the property (2)is satisfied.To prove (3), by Theorem 5, it is enough if we prove that any twoincomparable prime filters if (N,6pC) are co-maximal for all p ∈ P . For anypositive a and b, if a and b are incomparable in (N,6pC), then 〈a, b〉 /∈ πpand hence a and b have no upper bound and therefore a∨ b does not existin (N,6pC). Also, each progression in πp is a maximal chain in (N,6pC)and, for any a and b ∈ N , a and b are comparable if and only if 〈a, b〉 ∈ πp.Therefore (Z+,6pC) is a disjoint union of maximal chains. Thus, byTheorem 4, any two incomparable prime filters of (N,6pC) are co-maximal.Therefore, by Theorem 5, any two incomparable prime filters of (Z+,6C)are co-maximal. This proves (3).(4) follows from Theorem 6 and Theorem 7 and from the discussionmade above.To prove (5), let m and n ∈ Z+ such that m− <C n. By Theorem10 (3), we get that mn6C n. Let us writen = pb11 pb22 · · · pbrr and m = pa11 pa22 · · · parrwhere p1, p2, · · · , pr are distinct primes and each bi > 0 such that 0 6piCai 6piCbi. Since m 6= n, there exists i such that ai 6piCbi. Now, if aj 6piCbjfor some j 6= i, then the element k = pc11 pc22 · · · pcrr , wherecs ={as if s 6= i,bs if s = i.will be between m and n (that is, m <C k <C n) which is a contradiction.Therefore aj = bj for all j 6= i and hencenm= pai−bii .“adm-n1” — 2018/4/2 — 12:46 — page 154 — #156154 Characterization of regular convolutionsSince 〈ai, bi〉 ∈ pipi , there exists t > 0 such thatbi = ut and ai = vtfor some u and v with v < u. Also, vt, (v + 1)t, · · · , ut are all in the sameprogression. Since m− <C n, it follows that u = v + 1 and hencenm= pti.Since 0− < t in (N,6piC), we get that1− <C pt =nm6C n.This proves (5).Conversely suppose that C satisfies properties (1)–(5), since [pa)is prime filter of (Z+,6C) for all p ∈ P and a ∈ Z+, by Theorem 6,every proper filter of (N,6pC) is prime, for any p ∈ P . Since any twoincomparable prime filters of (Z+,6C) are co-maximal, by Theorem 8 andTheorem 5, we get that (Z+,6pC) is a disjoint union of maximal chains.Fix p ∈ P . ThenZ+ =∐i∈IYiwhere each Yi is a maximal chain in (Z+,6pC) such that, for any i 6= j ∈ I,Yi ∩ Yj = φ and each element of Yi is incomparable with each elementof Yj . Now, we shall prove that each Yi is an arithmetical progression(finite or infinite).Let i ∈ I. Since N is countable, Yi is at most countable. Also, since(N,6pC) satisfies the descending chain condition, we can expressYi = {a1− <C a2− <C a3− < . . . }By using induction on r, we shall prove that ar = ra1 for all r.Clearly, this is true for r = 1. Assume that r > 1 and as = sa1 for all1 6 s < r. Since (r − 1)a1 = ar−1− < ar in (N,6pC), we havepar−1− <C par in (Z+,6C)and hence, by condition (5),1− <C par−ar−1 6C par .Therefore, 0 6= ar − ar−1 6pCar and hence ar − ar−1 ∈ Yi (since ar ∈ Yi).Also, since 0− < ar − ar−1 in (N,6pC), we have ar − ar−1 = a1 andtherefore ar = ar−1 + a1 = (r − 1)a1 + a1 = ra1. Hence, for any prime p“adm-n1” — 2018/4/2 — 12:46 — page 155 — #157S. Sagi 155and a ∈ Z+,C(pa) = {1, pt, p2t, · · · , pst} and st = afor some positive integers t and s andpt ∈ C(p2t), p2t ∈ C(p3t), · · · , p(s−1)t ∈ C(pa).The other conditions given in Theorem 10 are clearly satisfied. Thus, byTheorem 10, C is a regular convolution.Theorem 13. Let C be a convolution, then the following conditions areequivalent to each other.(1) (Z+,6C) is a lattice.(2) (N,6pC) is a lattice for each p ∈ P .(3) (N,6pC) is a totally ordered set for each p ∈ P .(4) For any p ∈ P and a and b ∈ N , a 6pCb ⇐⇒ a 6 b.(5) For any n and m ∈ Z+, n 6C m ⇐⇒ n divides m.(6) C(n) = The set of positive divisors of n.Proof. Since C is regular, C ∼ {πp}p∈P .(1) =⇒ (2) follows from Theorem 3.(2) =⇒ (3) Let p ∈ P . Suppose that (N,6pC) is a lattice. If πp containstwo progressions, then choose an element a in one progression S and b inanother progression T in πp. Since a 6pCa∨ b and b 6pCa∨ b, a∨ b ∈ S∩T .A contradiction.Therefore πp contains only one progression, which must beN = {0 <pC1 <pC2 <pC3 <pC. . . }.Thus (N,6pC) is a totally ordered set.(3) =⇒ (4) It is trivial.(4) =⇒ (5) Let m and n ∈ Z+ and we write n =∏ri=1 Paii andm =∏ri=1 Pbii , where p1, p2, · · · , pr are distinct primes and ai, bi ∈ N .Now,n divides m ⇐⇒ ai 6 bi for all 1 6 i 6 r⇐⇒ ai 6pCbi for all 1 6 i 6 r⇐⇒ n 6C m.(5) =⇒ (6) For any n ∈ Z+,C(n) = {m ∈ Z+ | m 6C n} = {m ∈ Z+ | m divides n} = D(n).“adm-n1” — 2018/4/2 — 12:46 — page 156 — #158156 Characterization of regular convolutions(6) =⇒ (1) If C = D, then 6C=6D and, for any n,m ∈ Z+,n ∧m = gcd{n,m}and n ∨m = lcm{n,m} in (Z+,6C).The above Theorem implies that the Dirichlet’s convolution D is theonly regular convolution for which (Z+,6C) is a lattice.References[1] Cohen, E. Arithmetical functions associated with the unitary divisors of an integer.Math.Z.,74,66-80. 1960.[2] Narkiewicz, W. On a class of arithmetical convolutions. Collow.Math.,10,81-94.1963.[3] Sankar Sagi. Characterization of Prime Filters in (Z+,6C). International Journalof Pure and Engineering Mathematics, Vol.3, No III, 2015.[4] Sankar Sagi. Characterization of Prime Ideals in (Z+,6D). European Journal ofPure and Applied Mathematics,Vol. 8, No.1(15-25)2015.[5] Sankar Sagi. Co-maximal Filters in (Z+,6C). International Journal of Mathemat-ics and it’s Applications,Vol.3, Issue 4-C, 2015.[6] Sankar Sagi. Filters in (Z+,6C) and (N ,6pC). Journal of Algebra, Number Theory:Advances and Applications, Vol 11, No.2 (93-101)2014.[7] Sankar Sagi. Ideals in (Z+,6D). Algeba and Discrete Mathematics, Vol 16(2013),Number 1, pp 107-115.[8] Sankar Sagi. Irreducible elements in (Z+,6C). International Journal of Mathe-matics and it’s Applications, Vol.3, Issue 4-C, 2015.[9] Sankar Sagi. Lattice Structures on Z+ Induced by Convolutions. European Journalof Pure and Applied Mathematics, Vol. 4, No.4(424-434) 2011.[10] Sankar Sagi, Lattice Theory of Convolutions, Ph.D. Thesis, Andhra University,Waltair, Visakhapatnam, India. 2010.[11] Swamy, U.M., Rao, G.C., Sita Ramaiah, V. On a conjecture in a ring of arithmeticfunctions. Indian J.pure appl.Math.,14(12)1983.[12] Swamy, U.M., Sankar Sagi. Partial orders induced by convolutions. Internationaljournal of Mathematics and Soft Computing,Vol. 2, No.1(25-33) 2012.Contact informationSankar Sagi Assistant Professor of Mathematics, College ofApplied Sciences, Sohar, Sultanate of OmanE-Mail(s): sagi−sankar@yahoo.co.inReceived by the editors: 09.10.2015and in final form 03.02.2018.

Characterization of regular convolutions

http://dspace.nbuv.gov.ua/xmlui/bitstream/123456789/188354/1/12-Sagi.pdf

Characterization of regular convolutions

Abstract

Similar works

Full text

Available Versions

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)