Ideals in (Z⁺, ≤D)
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 s...
Saved in:
| Published in: | Algebra and Discrete Mathematics |
|---|---|
| Date: | 2013 |
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут прикладної математики і механіки НАН України
2013
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/152313 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Ideals in (Z⁺, ≤D) / S. Sagi // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 107–115. — Бібліогр.: 9 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-152313 |
|---|---|
| record_format |
dspace |
| spelling |
Sagi, S. 2019-06-09T17:20:35Z 2019-06-09T17:20:35Z 2013 Ideals in (Z⁺, ≤D) / S. Sagi // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 107–115. — Бібліогр.: 9 назв. — англ. 1726-3255 2010 MSC:06B10,11A99. https://nasplib.isofts.kiev.ua/handle/123456789/152313 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Ideals in (Z⁺, ≤D) Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Ideals in (Z⁺, ≤D) |
| spellingShingle |
Ideals in (Z⁺, ≤D) Sagi, S. |
| title_short |
Ideals in (Z⁺, ≤D) |
| title_full |
Ideals in (Z⁺, ≤D) |
| title_fullStr |
Ideals in (Z⁺, ≤D) |
| title_full_unstemmed |
Ideals in (Z⁺, ≤D) |
| title_sort |
ideals in (z⁺, ≤d) |
| author |
Sagi, S. |
| author_facet |
Sagi, S. |
| publishDate |
2013 |
| language |
English |
| container_title |
Algebra and Discrete Mathematics |
| publisher |
Інститут прикладної математики і механіки НАН України |
| format |
Article |
| description |
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.
|
| issn |
1726-3255 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/152313 |
| citation_txt |
Ideals in (Z⁺, ≤D) / S. Sagi // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 107–115. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT sagis idealsinzd |
| first_indexed |
2025-11-26T13:23:00Z |
| last_indexed |
2025-11-26T13:23:00Z |
| _version_ |
1850622622164844544 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 107 – 115
© Journal “Algebra and Discrete Mathematics”
Ideals in (Z+
, ≤D)
Sankar Sagi
Communicated by V. V. Kirichenko
Abstract. 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.
Introduction
A convolution is a mapping C : Z+ −→ P(Z+) such that C(n) is a
set of positive divisors on n, n ∈ C(n) and C(n) =
⋃
m∈C(n)
C(m), for any
n ∈ Z+, where Z+ is the set of all positive integers and P(Z+) is the set
of all subsets of Z+. Popular examples are the Dirichlet’s convolution D
and the Unitary convolution U defined respectively by
D(n) = The set of all positive divisors of n
and U(n)= {d / d|n and (d, n
d
) = 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]. It
is well known that the Dirichlet’s convolution induces the division order
on Z+ with respect to which Z+ becomes a distributive lattice, where, for
any a, b ∈ Z+, the greatest common divisor(GCD) and the least common
multiple(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 the
division order, the lattice Z+ satisfies the infinite join distributive law
given by
a ∨ (
∧
i∈I
bi) =
∧
i∈I
(a ∨ bi)
for any a ∈ Z+ and {bi}i∈I ⊆ Z+. In this paper, we discuss various
aspects of ideals in (Z+, ≤C). Actually a general convolution may not
induce a lattice structure on Z+. However , most of the convolutions
we are considering induce a meet semi lattice structure on Z+. For this
reason, we first consider a general semi lattice and study it’s ideals and
later extend these to (Z+, ≤D).
1. Preliminaries
Let us recall that a partial order on a non-empty set X is defined
as 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 that
a pair (X, ≤) is called a partially ordered set(poset) if X is a non-empty
set and ≤ is a partial order on X. For any A ⊆ X and x ∈ X, x is called
a 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 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 denoted by a1 ∧ a2 ∧ · · · ∧ an or
n
∧
i=1
ai (respectively
by a1 ∨ a2 ∨ · · · ∨ an or
n
∨
i=1
ai). A partially ordered set (X, ≤) is called a
meet 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 and
b ∈ X. A poset (X, ≤) is called a lattice if it is both a meet and join semi
lattice. Equivalently, lattice can also be defined as an algebraic system
S. Sagi 109
(X, ∧, ∨), where ∧ and ∨ are binary operations which are associative,
commutative and idempotent and satisfying the absorption laws, namely
a ∧ (a ∨ b) = a = a ∨ (a ∧ b) for all a, b ∈ X ; in this case the partial
order ≤ on X is such that a ∧ b and a ∨ b are respectively the glb and
lub of {a, b}. The algebraic operations ∧ and ∨ and the partial order ≤
are related by
a = a ∧ b ⇐⇒ a ≤ b ⇐⇒ a ∨ b = b.
Throughout the paper, Z+ and N denote the set of positive integers and
the set of non-negative integers respectively.
Definition 1. A mapping C : Z+ −→ P(Z+) is called a convolution if
the 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 define
m ≤ n if and only if m ∈ C(n)
Then ≤C is a partial order on Z+ and is called the partial order induced
by C on Z+. In fact, for any mapping C : Z+ −→ P(Z+) such that each
member of C(n) is a divisor of n, ≤C is a partial order on Z+ if and only
if C is a convolution, as defined above [6],[8].
Definition 3. Let C be a convolution and p a prime number. Define a
relation ≤p
C
on the set N of non-negative integers by
a ≤p
C
b if and only if pa ∈ C(pb)
for any a and b ∈ N .
It can be easily verified that ≤p
C
is a partial order on N , for each
prime 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 , ≤p
C
)
for each prime p.
(2) If (Z+, ≤C) is a lattice, then so is (N , ≤p
C
) for each prime p.
110 Ideals in (Z+, ≤D)
Now, we have the following examples from [9] in which the convolutions
induce meet semi lattice structures.
Example 1. Let D be the Dirichlet’s convolution defined by
D(n) = The set of all positive divisors of n.
Then ≤D is precisely the division order on Z+ and, for each prime p, ≤p
D
is the usual order on N . (Z+, ≤D) is known to be distributive lattice.
Example 2. Let U(n) be the Unitary convolution defined by
U(n) = {d ∈ D(n) | d and n
d
are relatively prime}.
Then (Z+, ≤U ) is a meet semilattice, but not a join semilattice.
Note that
U(pa) = {1, pa} for any 0 < a ∈ N .
Example 3. Let F2 be the square-free convolution defined by
F2(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. Note
that, for any prime p and a ∈ N ,
F2(pa) =
{1} if a = 0
{1, p} if a = 1
{1, p, pa} if a > 1
Example 4. For any k ∈ Z+, a positive integer d is said to be k-free if
pk does not divide d for any prime p. Let Fk(n) be the set of all k-free
divisors of n together with n. Then (Z+, ≤Fk
) is a meet semilattice but
not a join semi lattice.
2. Ideals in Semi lattices
Recall that most of the convolutions like Dirichlet’s convolution, Uni-
tary convolution and k-free convolution induce meet semi lattice structure
on Z+[9]. For this reason we consider a general meet semi lattice and
study it’s ideals. Throughout this section, unless otherwise stated, by a
semi lattice we mean a meet semi lattice only.
S. Sagi 111
Definition 4. Let (S, ∧) be a semi lattice. A non-empty subset I of S is
called 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 ≤ c
Theorem 2. Let a and b be elements of a meet semi lattice (S, ∧). Then
the 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 an
ideal 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 that
a ≤ x and b ≤ x
Therefore x is an upper bound of a and b. If y is any other upper bound
of 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 least
upper bound of a and b.
(3) =⇒ (1) : Let a ∨ b be the least upper bound of a and b. Then
a ≤ 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 that
a ≤ x and b ≤ x and hence a ∨ b ≤ x
so that a ∨ b ∈ I and (a ∨ b] ⊆ I. Thus (a ∨ b] is the smallest ideal of S
containing a and b.
Although the intersection of an arbitrary class of ideals need not be
an ideal, a finite intersection is always an ideal.
Theorem 3. Let (S, ∧) be a semi lattice and I(S) the set of all ideals of
S. 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 ≤ b
112 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 a
lattice and, in this case, a 7→ (a] is an embedding of the lattice S into the
lattice I(S) .
Proof. It is well known that the set I(S) of ideals of a lattice (S, ∧, ∨) is
again a lattice in which,
I ∧ J = I ∩ J
and 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 be
the least upper bound of (a] and (b] in I(S). Then I is the smallest ideal
containing a and b and hence by Theorem 3.3, a ∨ b exists in S. Therefore
S is a lattice.
For a lattice (L, ∧, ∨), any ideal of the semi lattice (L, ∧) turns out
to 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 lattice
structure on Z+ induced by the division ordering / and study the ideals
of Z+. The division ordering is precisely the partial ordering ≤D induced
by the Dirichlet’s convolution D.
First we observe that θ : (Z+, /) −→ (
∑
P
N , ≤) is an order isomor-
phism where θ is defined by
θ(a)(p) =The largest n in N such that pn divides a,for any a ∈ Z+ and p ∈
P and
∑
P
N = { f : P −→ N | f(p) = 0 for all but finite p ′s }. Here
P stands for the set of primes and N stands for the set of non-negative
integers.
Definition 5. Adjoin an external element ∞ to N and extend the usual
ordering ≤ on N to N ∪ {∞} by defining a < ∞ for all a ∈ N . We shall
denote N ∪ {∞} together with this extended usual order by N ∞ .
S. Sagi 113
Theorem 5. Let α : P −→ N ∞ be a mapping and define
Iα = { 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 form
Iα for some mapping α : P −→ N ∞
Proof. Since no prime divides the integer 1, we get that
θ(1)(p) = 0 ≤ α(p) for all p ∈ P and hence 1 ∈ Iα. Therefore Iα is a
non-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 ∈ Iα
and
m ≤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 ∞
by
α(p) = Sup{ θ(n)(p) | n ∈ I } for any p ∈ P
Note 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 ∈ Iα
Therefore I ⊆ Iα.
On the other hand, suppose n ∈ Iα. Then θ(n)(p) ≤ α(p) for all
p ∈ P. Since θ(n) ∈
∑
P
N , |θ(n)| is finite. If |θ(n)| = φ, then n = 1 ∈ I.
Suppose |θ(n)| is non-empty. Let |θ(n)| = { p1, p2 · · · , pr }. Then
θ(n)(p) = 0 for all p 6= pi, 1 ≤ i ≤ r and θ(n)(pi) ∈ N . Now, for
each 1 ≤ i ≤ r, θ(n)(pi) ≤ α(pi) = Sup{ θ(m)(pi) | m ∈ I } and
hence there exists mi ∈ I such that θ(n)(pi) ≤ θ(m)(pi). Now, put
m = m1 ∨ m2 ∨ · · · ∨ mr, then m ∈ I and
θ(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) for
114 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 ∞ , define
α ≤ 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. If
β(p) = ∞ or α(p) = 0, trivially α(p) ≤ β(p). Therefore, we can assume
that β(p) < ∞ and α(p) > 0.
Consider n = pβ(p)+1. Then
θ(n)(p) = β(p) + 1 � β(p).
and hence n /∈ Iβ . Since Iα ⊆ Iβ , n /∈ Iα and therefore θ(n)(q) � α(q) for
some q ∈ P. But θ(n)(q) = 0 for all q 6= p. Thus
β(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→ Iα
is 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 115
References
[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 arithmetic
functions, Indian J.pure appl.Math., 14(12),1519-1530.1983.
[8] U.M. Swamy and Sagi Sankar, Partial Orders induced by Convolutions, International
Journal 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 information
Sankar Sagi Assistant Professor of Mathematics, College of
Applied Sciences, Sohar, Sultanate of Oman
E-Mail: sagi−sankar@yahoo.co.in
Received by the editors: 17.12.2011
and in final form 27.03.2013.
|