Characterization of regular convolutions
In this paper, we present a characterization of regular convolution.
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irk-123456789-1883542023-02-24T01:27:26Z Characterization of regular convolutions Sagi S. In this paper, we present a characterization of regular convolution. 2018 Article Characterization of regular convolutions / S. Sagi // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 147-156. — Бібліогр.: 12 назв. — англ. 1726-3255 2010 MSC: 06B10, 11A99. http://dspace.nbuv.gov.ua/handle/123456789/188354 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper, we present a characterization of regular convolution. |
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Sagi S. |
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Sagi S. Characterization of regular convolutions Algebra and Discrete Mathematics |
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Sagi S. |
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Sagi S. |
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Characterization of regular convolutions |
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Characterization of regular convolutions |
| title_full |
Characterization of regular convolutions |
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Characterization of regular convolutions |
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Characterization of regular convolutions |
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characterization of regular convolutions |
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Інститут прикладної математики і механіки НАН України |
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2018 |
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Characterization of regular convolutions / S. Sagi // Algebra and Discrete Mathematics. — 2018. — Vol. 25, № 1. — С. 147-156. — Бібліогр.: 12 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT sagis characterizationofregularconvolutions |
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2025-07-16T10:22:42Z |
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1837798642140839936 |
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“adm-n1” — 2018/4/2 — 12:46 — page 147 — #149
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 25 (2018). Number 1, pp. 147–156
c© Journal “Algebra and Discrete Mathematics”
Characterization of regular convolutions
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,
for any n ∈ Z+, each member of C(n) is a divisor of n. If D(n) is
the set of all divisors of n, for any n, then D is called the Dirichlet’s
convolution [2]. If U(n) is the set of all Unitary(square free) divisors
of n, for any n, then U is called unitary(square free) convolution.
Corresponding to any general convolution C, we can define a binary
relation 6C on Z+ by ‘m 6C n if and only if m ∈ C(n)’. In this
paper, we present a characterization of regular convolution.
Introduction
A convolution is a mapping C of the set Z+ of positive integers into
the set P(Z+) of subsets of Z+ such that, for any n ∈ Z+, C(n) is a
nonempty set of divisors of n. If C(n) is the set of all divisors of n, for
each n ∈ Z+, then C is the classical Dirichlet convolution [2]. If
C(n) = {d / d|n and (d,
n
d
) = 1},
then C is the Unitary convolution [1]. As another example if
C(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, we
can define a binary relation 6C in a natural way by
m 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 — #150
148 Characterization of regular convolutions
6C is a partial order on Z+ and is called partial order induced by the
convolution C [11], [12]. W. Narkiewicz [2] first proposed the concept of
a regular convolution, and in this paper we present a lattice theoretic
characterization of regular convolution and prove that the Dirichlet’s con-
volution is the unique regular convolution that induces a lattice structure
on (Z+,6C).
1. Preliminaries
Let 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 a
partial order on X.
For any A ⊆ X and x ∈ X, x is called a lower(upper) bound of A if
x 6 a (respectively a 6 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,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 lattice
if a ∨ b (=lub{a, b}) exists for all a and b ∈ X. A poset (X,6) is called
a lattice if it is both a meet and join semi lattice. Equivalently, lattice
can also be defined as an algebraic system (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 6 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 6 are related by
a = 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 — #151
S. Sagi 149
Corollary 1 ([12]). The binary relation 6C induced by convolution C
on 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 be
binary relations on X and Y respectively. A bijection f : X → Y is said
to be a relation isomorphism of (X,R) into (Y, S) if, for any elements a
and 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 by
θ(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) is
a lattice if and only if (N,6p
C
) is a lattice for each prime p.
Now we state the following theorems on co-maximality and prime
filters.
Theorem 4 ([5]). Let (S,∧) be any meet semi lattice with smallest element
0 satisfying the descending chain condition. Also, suppose that every proper
filter 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 filters
of (Z+,6C) are co-maximal if and only if any two incomparable prime
filters of (N,6p
C
) are co-maximal, for each p ∈ P .
Theorem 6 ([3]). Let p be a prime number. Then every proper filter in
(N,6p
C
) is prime if and only if [pa) is a prime filter in (Z+,6C) for all
n > 0.
Theorem 7 ([3]). A filter F of (Z+,6C) is prime if and only if there
exists unique p ∈ P such that F p is a prime filter of (N,6p
C
) and F q = N
for all q 6= p in P and, in this case,
F = {n ∈ Z+ | θ(n)(p) ∈ F p}.
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150 Characterization of regular convolutions
Theorem 8 ([3]). Let F be a filter of (Z+,6C). Then F = [pa) for
some 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 arithmetical
functions is denoted by A.
The following is a routine verification using the properties of addition
and 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 arithmetical
functions and (A,+, ·) is a commutative ring with unity in which the
constant map 0̄ and 1̄ are the zero element and unity element respectively.
Definition 3. Let C be a convolution and f and g arithmetical functions
and C be the field of complex numbers. Define fCg : Z+ → C by
(fCg)(n) =
∑
d∈C(n)
f(d)g(
n
d
).
We can consider C as a binary operation, as defined above, on the
set A of arithmetical functions. W.Narkiewicz proposed the following
definition.
Definition 4 ([2]). A convolution C is called regular if the following are
satisfied.
(1) (A,+, C) is a commutative ring with unity, where + is the point-wise
addition. This ring will be denoted by AC .
(2) If f and g are multiplicative arithmetical functions, then so is the
product 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 a
unit in the ring AC .
It can be easily verified that the arithmetical function e, defined by
e(n) =
{
1 if n = 1,
0 if n > 1
is the unity (the identity element with respect to the binary operation C).
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S. Sagi 151
W. Narkiewicz proved the following two theorems.
Theorem 10 ([2]). A convolution C is regular if and only if the following
conditions 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 m
d
∈ C(n
d
).
(3) d ∈ C(n) ⇒ n
d
∈ 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 of
non-negative integers into arithmetic progressions (finite or infinite) each
containing 0 and no two progressions belonging to same decomposition
have 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 are
distinct primes and a1, a2, · · · , ar ∈ N , define
C(n) = {pb11 pb22 · · · pbrr | bi 6 ai, and bi and ai belong to the same
progression in πpi .}
Then C is a regular convolution and, conversely every regular convolution
can be obtained in this way.
From the above theorems, it is clear that any regular convolution C
is uniquely determined by a sequence {πp}p∈P of decompositions of N
into arithmetical progressions (finite or infinite) and we denote this by
expression 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 is
no c ∈ X such that a < c < b. This is denoted by a− < b.
We note that θ : Z+ →
∑
P N defined by
θ(a)(p) = the largest n in N such that pn divides a,
for any a ∈ Z+ and p ∈ P
is a bijection.
2. Main results
In the following two theorems, we prove that any regular convolution C
gives a meet semi lattice structure on (Z+,6C) and the convolution C is
“adm-n1” — 2018/4/2 — 12:46 — page 152 — #154
152 Characterization of regular convolutions
completely characterized by certain lattice theoretic properties of (Z+,6C).
In particular Dirichlet’s convolution is the only regular convolution C which
gives a lattice structure on (Z+,6C).
Theorem 12. Let C be a convolution and 6C the relation on Z+ induced
by C. Then C is a regular convolution if and only if the following properties
are satisfied.
(1) θ : (Z+,6C) →
∑
p∈P (N,6p
C
) 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 some
p ∈ P and a ∈ Z+.
(5) For any m and n ∈ Z+, m− <C n =⇒ 1− <C
n
m
6C 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 arithmetic
progressions (finite or infinite) in which each progression contains 0 and
no positive integer belongs to two distinct progressions. For any a, b ∈ N
and 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 partial
order on Z+. Since C is multiplicative, it follows from Theorem 2 that
θ : Z+ →
∑
P N is an order isomorphism. Therefore the property (1) is
satisfied. For simplicity and convenience, we shall write n̄ for θ(n). For
each n ∈ Z+, n̄ is the element in the direct sum
∑
P N defined by
n̄(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,6p
C
), where for
each p ∈ P , 6p
C
is the partial order on N defined by
a 6
p
C
b if and only if pa ∈ C(pb).
For any m and n ∈ Z+, let m ∧ n be the element in Z+ defined by
m ∧ 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, then
m(p) 6p
C
n(p) or n(p) 6p
C
m(p)
“adm-n1” — 2018/4/2 — 12:46 — page 153 — #155
S. Sagi 153
and hence m ∧ n(p) 6 m(p) and n(p) for all p ∈ P . Therefore m ∧ n is a
lower bound of m and n in (Z+,6C). Let k be any other lower bound of
m and n. For any p ∈ P , if 〈m(p), n(p)〉 ∈ πp, then, since
k(p) 6p
C
m(p) and k(p) 6p
C
n(p),
we have
k(p) 6p
C
m ∧ n(p).
If 〈m(p), n(p)〉 /∈ πp, then
k(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 two
incomparable prime filters if (N,6p
C
) are co-maximal for all p ∈ P . For any
positive a and b, if a and b are incomparable in (N,6p
C
), then 〈a, b〉 /∈ πp
and hence a and b have no upper bound and therefore a∨ b does not exist
in (N,6p
C
). Also, each progression in πp is a maximal chain in (N,6p
C
)
and, for any a and b ∈ N , a and b are comparable if and only if 〈a, b〉 ∈ πp.
Therefore (Z+,6p
C
) is a disjoint union of maximal chains. Thus, by
Theorem 4, any two incomparable prime filters of (N,6p
C
) 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 discussion
made above.
To prove (5), let m and n ∈ Z+ such that m− <C n. By Theorem
10 (3), we get that m
n
6C n. Let us write
n = pb11 pb22 · · · pbrr and m = pa11 pa22 · · · parr
where p1, p2, · · · , pr are distinct primes and each bi > 0 such that 0 6
pi
C
ai 6
pi
C
bi. Since m 6= n, there exists i such that ai 6
pi
C
bi. Now, if aj 6
pi
C
bj
for some j 6= i, then the element k = pc11 pc22 · · · pcrr , where
cs =
{
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 hence n
m
= pai−bi
i .
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154 Characterization of regular convolutions
Since 〈ai, bi〉 ∈ pipi , there exists t > 0 such that
bi = ut and ai = vt
for some u and v with v < u. Also, vt, (v + 1)t, · · · , ut are all in the same
progression. Since m− <C n, it follows that u = v + 1 and hence n
m
= pti.
Since 0− < t in (N,6pi
C
), we get that
1− <C pt =
n
m
6C 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,6p
C
) is prime, for any p ∈ P . Since any two
incomparable prime filters of (Z+,6C) are co-maximal, by Theorem 8 and
Theorem 5, we get that (Z+,6p
C
) is a disjoint union of maximal chains.
Fix p ∈ P . Then
Z+ =
∐
i∈I
Yi
where each Yi is a maximal chain in (Z+,6p
C
) such that, for any i 6= j ∈ I,
Yi ∩ Yj = φ and each element of Yi is incomparable with each element
of 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,6p
C
) satisfies the descending chain condition, we can express
Yi = {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 all
1 6 s < r. Since (r − 1)a1 = ar−1− < ar in (N,6p
C
), we have
par−1− <C par in (Z+,6C)
and hence, by condition (5),
1− <C par−ar−1 6C par .
Therefore, 0 6= ar − ar−1 6
p
C
ar and hence ar − ar−1 ∈ Yi (since ar ∈ Yi).
Also, since 0− < ar − ar−1 in (N,6p
C
), we have ar − ar−1 = a1 and
therefore ar = ar−1 + a1 = (r − 1)a1 + a1 = ra1. Hence, for any prime p
“adm-n1” — 2018/4/2 — 12:46 — page 155 — #157
S. Sagi 155
and a ∈ Z+,
C(pa) = {1, pt, p2t, · · · , pst} and st = a
for some positive integers t and s and
pt ∈ C(p2t), p2t ∈ C(p3t), · · · , p(s−1)t ∈ C(pa).
The other conditions given in Theorem 10 are clearly satisfied. Thus, by
Theorem 10, C is a regular convolution.
Theorem 13. Let C be a convolution, then the following conditions are
equivalent to each other.
(1) (Z+,6C) is a lattice.
(2) (N,6p
C
) is a lattice for each p ∈ P .
(3) (N,6p
C
) is a totally ordered set for each p ∈ P .
(4) For any p ∈ P and a and b ∈ N , a 6
p
C
b ⇐⇒ 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,6p
C
) is a lattice. If πp contains
two progressions, then choose an element a in one progression S and b in
another progression T in πp. Since a 6
p
C
a∨ b and b 6p
C
a∨ b, a∨ b ∈ S∩T .
A contradiction.
Therefore πp contains only one progression, which must be
N = {0 <p
C
1 <p
C
2 <p
C
3 <p
C
. . . }.
Thus (N,6p
C
) is a totally ordered set.
(3) =⇒ (4) It is trivial.
(4) =⇒ (5) Let m and n ∈ Z+ and we write n =
∏r
i=1 P
ai
i and
m =
∏r
i=1 P
bi
i , 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 6
p
C
bi 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).
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156 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 the
only regular convolution for which (Z+,6C) is a lattice.
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[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 arithmetic
functions. Indian J.pure appl.Math.,14(12)1983.
[12] Swamy, U.M., Sankar Sagi. Partial orders induced by convolutions. International
journal of Mathematics and Soft Computing,Vol. 2, No.1(25-33) 2012.
Contact information
Sankar Sagi Assistant Professor of Mathematics, College of
Applied Sciences, Sohar, Sultanate of Oman
E-Mail(s): sagi−sankar@yahoo.co.in
Received by the editors: 09.10.2015
and in final form 03.02.2018.
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