A convolution is a mapping C of the set Z⁺ of positive integers into the set P(Z⁺) of all subsets of Z⁺ such that every member of C(n) is a divisor of n. If for any n, D(n) is the set of all positive divisors of n, then D is called the Dirichlet's convolution. It is well known that Z⁺ has the structure of a distributive lattice with respect to the division order. Corresponding to any general convolution C, one can define a binary relation ≤C on Z⁺ by 'm ≤ C n if and only if m ∈ C(n) '. A general convolution may not induce a lattice on Z⁺. However most of the convolutions induce a meet semi lattice structure on Z⁺. In this paper we consider a general meet semi lattice and study it's ideals and extend these to (Z⁺, ≤D), where D is the Dirichlet's convolution

Sagi, S.

Algebra and Discrete Mathematics

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

Algebra and Discrete Mathematics RESEARCH ARTICLEVolume 16 (2013). Number 1. pp. 107 – 115© Journal “Algebra and Discrete Mathematics”Ideals in (Z+,≤D)Sankar 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 thatevery member of C(n) is a divisor of n. If for any n,D(n) is the set ofall positive divisors of n , then D is called the Dirichlet’s convolution.It is well known that Z+ has the structure of a distributive latticewith respect to the division order. Corresponding to any generalconvolution C, one can define a binary relation ≤C on Z+ by ‘m ≤C n if and only if m ∈ C(n) ’ . A general convolution may notinduce a lattice on Z+ . However most of the convolutions inducea meet semi lattice structure on Z+ .In this paper we consider ageneral meet semi lattice and study it’s ideals and extend these to(Z+,≤D) , where D is the Dirichlet’s convolution.IntroductionA convolution is a mapping C : Z+ −→ P(Z+) such that C(n) is aset of positive divisors on n, n ∈ C(n) and C(n) =⋃m∈C(n)C(m), for anyn ∈ Z+, where Z+ is the set of all positive integers and P(Z+) is the setof all subsets of Z+. Popular examples are the Dirichlet’s convolution Dand the Unitary convolution U defined respectively byD(n) = The set of all positive divisors of nand U(n)= {d / d|n and (d, nd) = 1}2010 MSC: 06B10,11A99.Key words and phrases: Partial Order,Lattice,Semi Lattice,Convolution,Ideal.108 Ideals in (Z+,≤D)for any n ∈ Z+. If C is a convolution, then the binary relation ≤C on Z+,defined by,m ≤C n if and only if m ∈ C(n) ,is a partial order on Z+ and is called the partial order induced by C [7]. Itis well known that the Dirichlet’s convolution induces the division orderon Z+ with respect to which Z+ becomes a distributive lattice, where, forany a, b ∈ Z+, the greatest common divisor(GCD) and the least commonmultiple(LCM) of a and b are respectively the greatest lower bound(glb)and the least upper bound(lub) of a and b . In fact, with respect to thedivision order, the lattice Z+ satisfies the infinite join distributive lawgiven bya ∨ (∧i∈Ibi) =∧i∈I(a ∨ bi)for any a ∈ Z+ and {bi}i∈I ⊆ Z+. In this paper, we discuss variousaspects of ideals in (Z+,≤C). Actually a general convolution may notinduce a lattice structure on Z+. However , most of the convolutionswe are considering induce a meet semi lattice structure on Z+. For thisreason, we first consider a general semi lattice and study it’s ideals andlater extend these to (Z+,≤D).1. PreliminariesLet us recall that a partial order on a non-empty set X is definedas a binary relation ≤ on X which is reflexive (a ≤ a), transitive (a ≤b, b ≤ c =⇒ a ≤ c) and antisymmetric (a ≤ b, b ≤ a =⇒ a = b) and thata pair (X,≤) is called a partially ordered set(poset) if X is a non-emptyset and ≤ is a partial order on X. For any A ⊆ X and x ∈ X, x is calleda lower(upper) bound of A if x ≤ a(respectively a ≤ x) for all a ∈ A.We have the usual notations of the greatest lower bound(glb) and leastupper bound(lub) of A in X. If A is a finite subset {a1, a2, · · · , an}, theglb of A(lub of A) is denoted by a1 ∧ a2 ∧ · · · ∧ an orn∧i=1ai (respectivelyby a1 ∨ a2 ∨ · · · ∨ an orn∨i=1ai). A partially ordered set (X,≤) is called ameet semi lattice if a ∧ b (=glb{a, b}) exists for all a and b ∈ X. (X,≤)is called a join semi lattice if a ∨ b (=lub{a, b}) exists for all a andb ∈ X. A poset (X,≤) is called a lattice if it is both a meet and join semilattice. Equivalently, lattice can also be defined as an algebraic systemS. Sagi 109(X,∧,∨), where ∧ and ∨ are binary operations which are associative,commutative and idempotent and satisfying the absorption laws, namelya ∧ (a ∨ b) = a = a ∨ (a ∧ b) for all a, b ∈ X ; in this case the partialorder ≤ on X is such that a ∧ b and a ∨ b are respectively the glb andlub of {a, b}. The algebraic operations ∧ and ∨ and the partial order ≤are related bya = a ∧ b ⇐⇒ a ≤ b ⇐⇒ a ∨ b = b.Throughout the paper, Z+ and N denote the set of positive integers andthe set of non-negative integers respectively.Definition 1. A mapping C : Z+ −→ P(Z+) is called a convolution ifthe following are satisfied for any n ∈ Z+.(1) C(n) is a set of positive divisors of n(2) n ∈ C(n)(3) C(n) =⋃m∈C(n)C(m).Definition 2. For any convolution C and m and n ∈ Z+, we definem ≤ n if and only if m ∈ C(n)Then ≤C is a partial order on Z+ and is called the partial order inducedby C on Z+. In fact, for any mapping C : Z+ −→ P(Z+) such that eachmember of C(n) is a divisor of n, ≤C is a partial order on Z+ if and onlyif C is a convolution, as defined above [6],[8].Definition 3. Let C be a convolution and p a prime number. Define arelation ≤pCon the set N of non-negative integers bya ≤pCb if and only if pa ∈ C(pb)for any a and b ∈ N .It can be easily verified that ≤pCis a partial order on N , for eachprime p. The following is a direct verification.Theorem 1. Let C be a convolution.(1) If (Z+,≤C) is a meet(join) semilattice, then so is (N ,≤pC)for each prime p.(2) If (Z+,≤C) is a lattice, then so is (N ,≤pC) for each prime p.110 Ideals in (Z+,≤D)Now, we have the following examples from [9] in which the convolutionsinduce meet semi lattice structures.Example 1. Let D be the Dirichlet’s convolution defined byD(n) = The set of all positive divisors of n.Then ≤D is precisely the division order on Z+ and, for each prime p, ≤pDis the usual order on N . (Z+,≤D) is known to be distributive lattice.Example 2. Let U(n) be the Unitary convolution defined byU(n) = {d ∈ D(n) | d and ndare relatively prime}.Then (Z+,≤U ) is a meet semilattice, but not a join semilattice.Note thatU(pa) = {1, pa} for any 0 < a ∈ N .Example 3. Let F2 be the square-free convolution defined byF2(n) = {n} ∪ {d ∈ D(n) | p2 does not divide n for any prime p}.Then (Z+,≤F2) is a meet semilattice but not a join semilattice. Notethat, for any prime p and a ∈ N ,F2(pa) ={1} if a = 0{1, p} if a = 1{1, p, pa} if a > 1Example 4. For any k ∈ Z+, a positive integer d is said to be k-free ifpk does not divide d for any prime p. Let Fk(n) be the set of all k-freedivisors of n together with n. Then (Z+,≤Fk) is a meet semilattice butnot a join semi lattice.2. Ideals in Semi latticesRecall that most of the convolutions like Dirichlet’s convolution, Uni-tary convolution and k-free convolution induce meet semi lattice structureon Z+[9]. For this reason we consider a general meet semi lattice andstudy it’s ideals. Throughout this section, unless otherwise stated, by asemi lattice we mean a meet semi lattice only.S. Sagi 111Definition 4. Let (S,∧) be a semi lattice. A non-empty subset I of S iscalled an ideal of S if the following are satisfied(1) x ∈ S and x ≤ a ∈ I =⇒ x ∈ I(2) For any a and b ∈ I, there exists c ∈ I such that a ≤ c and b ≤ cTheorem 2. Let a and b be elements of a meet semi lattice (S,∧). Thenthe following are equivalent to each other.(1) There exists smallest ideal of S containing a and b.(2) The intersection of all ideals of S containing a and b is again anideal of S.(3) a and b have least upper bound in S.Proof. (1)⇐⇒ (2) is trivial.(1) =⇒ (3) : Let I be the smallest ideal of S containing a and b. Then,there exists x ∈ I such thata ≤ x and b ≤ xTherefore x is an upper bound of a and b. If y is any other upper boundof a and b, then (y] is an ideal of S containing a and b and hence I ⊆ (y].Since x ∈ I, we get that x ∈ (y] and therefore x ≤ y. Thus x is the leastupper bound of a and b.(3) =⇒ (1) : Let a ∨ b be the least upper bound of a and b. Thena ≤ a ∨ b and b ≤ a ∨ b and hence (a ∨ b] is an ideal containing a and b.If I is any ideal containing a and b, then there exists x ∈ I such thata ≤ x and b ≤ x and hence a ∨ b ≤ xso that a ∨ b ∈ I and (a ∨ b] ⊆ I. Thus (a ∨ b] is the smallest ideal of Scontaining a and b.Although the intersection of an arbitrary class of ideals need not bean ideal, a finite intersection is always an ideal.Theorem 3. Let (S,∧) be a semi lattice and I(S) the set of all ideals ofS. Then (I(S),∩) is a semilattice and a 7→ (a] is an embedding of (S,∧)onto (I(S),∩).Proof. By the above theorem, it follows that (I(S),∩) is a semi lat-tice.Also, for any a and b in S, we have(a] ∩ (b] = (a ∧ b]and (a] ⊆ (b] ⇐⇒ a ∈ (b] ⇐⇒ a ≤ b112 Ideals in (Z+,≤D)Therefore a 7→ (a] is an embedding of S into I(S).Theorem 4. A semi lattice (S,∧) is a lattice if and only if I(S) is alattice and, in this case, a 7→ (a] is an embedding of the lattice S into thelattice I(S) .Proof. It is well known that the set I(S) of ideals of a lattice (S,∧,∨) isagain a lattice in which,I ∧ J = I ∩ Jand I ∨ J = { x ∈ S | x ≤ a ∧ b, for some a ∈ I and b ∈ J }for any ideals I and J , in this case,(a] ∨ (b] = (a ∨ b]for any a and b in S, so that a 7→ (a] is an embedding of lattices.Conversely, suppose that I(S) is a lattice. Let a and b ∈ S and I bethe least upper bound of (a] and (b] in I(S). Then I is the smallest idealcontaining a and b and hence by Theorem 3.3, a∨ b exists in S. ThereforeS is a lattice.For a lattice (L,∧,∨), any ideal of the semi lattice (L,∧) turns outto be the usual ideal of the lattice (L,∧,∨).3. Ideals in (Z+,≤D)Now we shall turn our attention to the particular case of the latticestructure on Z+ induced by the division ordering / and study the idealsof Z+. The division ordering is precisely the partial ordering ≤D inducedby the Dirichlet’s convolution D.First we observe that θ : (Z+, /) −→ (∑PN ,≤) is an order isomor-phism where θ is defined byθ(a)(p) =The largest n inN such that pn divides a,for any a ∈ Z+ and p ∈P and∑PN = { f : P −→ N | f(p) = 0 for all but finite p ′s }. HereP stands for the set of primes and N stands for the set of non-negativeintegers.Definition 5. Adjoin an external element ∞ to N and extend the usualordering ≤ on N to N ∪ {∞} by defining a <∞ for all a ∈ N . We shalldenote N ∪ {∞} together with this extended usual order by N∞ .S. Sagi 113Theorem 5. Let α : P −→ N∞ be a mapping and defineIα = { n ∈ Z+ | θ(n)(p) ≤ α(p) for all p ∈ P}Then Iα is an ideal of (Z+, /) and every ideal of (Z+, /) is of the formIα for some mapping α : P −→ N∞Proof. Since no prime divides the integer 1, we get thatθ(1)(p) = 0 ≤ α(p) for all p ∈ P and hence 1 ∈ Iα. Therefore Iα is anon-empty subset of Z+.m and n ∈ Iα =⇒ θ(m)(p) ≤ α(p) and θ(n)(p) ≤ α(p) for all p ∈ P=⇒ θ(m∨ n)(p) = Max { θ(m)(p), θ(n)(p) } ≤ α(p)for all p ∈ P=⇒ m ∨ n ∈ Iαandm ≤D n ∈ Iα =⇒ θ(m)(p) ≤ θ(n)(p) ≤ α(p) for all p ∈ P=⇒ θ(m)(p) ≤ α(p) for all p ∈ P=⇒ m ∈ Iα.Thus Iα is an ideal of (Z+, /).Conversely suppose that I is any ideal of (Z+, /). Define α : P −→ N∞byα(p) = Sup{ θ(n)(p) | n ∈ I } for any p ∈ PNote that α(p) is either a non-negative integer or ∞, for any p ∈ P.Therefore α is a mapping of P into N∞.n ∈ I =⇒ θ(n)(p) ≤ α(p) for all p ∈ P=⇒ n ∈ IαTherefore I ⊆ Iα.On the other hand, suppose n ∈ Iα. Then θ(n)(p) ≤ α(p) for allp ∈ P. Since θ(n) ∈∑PN , |θ(n)| is finite. If |θ(n)| = φ, then n = 1 ∈ I.Suppose |θ(n)| is non-empty. Let |θ(n)| = { p1, p2 · · · , pr }. Thenθ(n)(p) = 0 for all p 6= pi, 1 ≤ i ≤ r and θ(n)(pi) ∈ N . Now, foreach 1 ≤ i ≤ r, θ(n)(pi) ≤ α(pi) = Sup{ θ(m)(pi) | m ∈ I } andhence there exists mi ∈ I such that θ(n)(pi) ≤ θ(m)(pi). Now, putm = m1 ∨m2 ∨ · · · ∨mr, then m ∈ I andθ(n)(pi) ≤ Max.{ θ(m1)(pi), · · · , θ(mi)(pi) } = θ(m)(pi) for all 1 ≤ i ≤ r.Also, since θ(n)(p) = 0 for all p 6= pi, we get that θ(n)(p) ≤ θ(m)(p) for114 Ideals in (Z+,≤D)all p ∈ P so that n ≤D m ∈ I and therefore n ∈ I. Therefore Iα ⊆ I.Thus I = Iα.Note that, if α is the constant map 0 defined by α(p) = 0 for all p ∈ P,then Iα = {1} and that , if α is the constant map ∞, then Iα = Z+.Definition 6. For any mappings α and β from P into N∞ , defineα ≤ β if and only if α(p) ≤ β(p) for all p ∈ P.Thus ≤ is a partial order on (N∞)P .Theorem 6. The map α 7→ Iα is an order isomorphism of the poset((N∞)P ,≤), onto the poset (I(Z+),⊆) of all ideals of (Z+, /).Proof.Let α and β : P 7→ N∞ be any mappings. Clearly, α ≤ β ⇒ Iα ⊆ Iβ .On the other hand, suppose that Iα ⊆ Iβ . We shall prove that α(p) ≤β(p) for all p ∈ P so that α ≤ β. To prove this, let us fix p ∈ P. Ifβ(p) =∞ or α(p) = 0, trivially α(p) ≤ β(p). Therefore, we can assumethat β(p) <∞ and α(p) > 0.Consider n = pβ(p)+1. Thenθ(n)(p) = β(p) + 1  β(p).and hence n /∈ Iβ . Since Iα ⊆ Iβ , n /∈ Iα and therefore θ(n)(q)  α(q) forsome q ∈ P. But θ(n)(q) = 0 for all q 6= p. Thusβ(p) + 1 = θ(n)(p)  α(p)α(p) < β(p) + 1.Therefore α(p) ≤ β(p). This is true for all p ∈ P. Thus α ≤ β. Also α 7→ Iαis a surjection. Thus α 7→ Iα is an order isomorphism of ((N∞)P ,≤),onto (I(Z+),⊆).Corollary 1. For any α and β : P → N∞,Iα ∩ Iβ = Iα ∧ β.and Iα ∪ Iβ = Iα ∨ β.where α ∧ β and α ∨ β are point-wise g.l.b and l.u.b of α and β.S. Sagi 115References[1] G. Birkhoff, Lattice Theory, American Mathematical Society, 1967.[2] E. Cohen, Arithmetical functions associated with the unitary divisors of an integer,Math.Z.,74,66-80.1960.[3] B.A. Davey and H.A. Priestly, Introduction to lattices and order, Cambridge Uni-versity Press,2002.[4] G.A Gratzer, General Lattice Theory, Birkhauser,1971.[5] W. Narkiewicz,On a class of arithmetical convolutions, Colloq. Math., 10,81-94.1963.[6] Sagi Sankar , Lattice Theory of Convolutions, Ph.D. Thesis, Andhra University,Waltair, Visakhapatnam, India, 2010.[7] U.M. Swamy,G.C. Rao, and V. Sita Ramaiah,On a conjecture in a ring of arithmeticfunctions, Indian J.pure appl.Math., 14(12),1519-1530.1983.[8] U.M. Swamy and Sagi Sankar, Partial Orders induced by Convolutions, InternationalJournal of Mathematics and Soft Computing, 2(1), 2011.[9] U.M. Swamy and Sagi Sankar, Lattice Structures on Z+ induced by convolutions,Eur. J. Pure Appl. Math., 4(4), 424-434. 2011.Contact informationSankar Sagi Assistant Professor of Mathematics, College ofApplied Sciences, Sohar, Sultanate of OmanE-Mail: sagi−sankar@yahoo.co.inReceived by the editors: 17.12.2011and in final form 27.03.2013.

Ideals in (Z⁺, ≤D)

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Ideals in (Z⁺, ≤D)

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