arXiv:1701.00058v1 [] 31 Dec 2016

THREE FAMILIES OF DENSE PUISEUX MONOIDS
FELIX GOTTI, MARLY GOTTI, AND HAROLD POLO
Abstract. A dense Puiseux monoid is an additive submonoid of Q≥0 whose topological closure is R≥0 . It follows immediately that every Puiseux monoid failing to be
dense is atomic. However, the atomic structure of dense Puiseux monoids is significantly complex. Dense Puiseux monoids can be antimatter, atomic, or anything in
between, meaning non-atomic with finitely or countably many atoms. In the first part
of this paper, we construct infinitely many non-isomorphic atomic Puiseux monoids
whose sets of atoms are dense in R≥0 . We dedicate the rest of the paper to study
the atomic structure of three families of dense Puiseux monoids: k-primary, p-adic,
and multiplicatively cyclic. In particular, we characterize an antimatter subfamily of
k-primary Puiseux monoids and a hereditarily atomic subfamily of multiplicatively
cyclic Puiseux monoids.

1. Introduction
A Puiseux monoid is an additive submonoid of the nonnegative rational numbers.
The atomic structure of Puiseux monoids has been recently studied in [6] and [8].
Albeit a natural generalization of numerical semigroups, Puiseux monoids exhibit a
complex and striking atomic constitution. For instance, the Puiseux monoid M in
Example 2.3 does not have any atoms. Furthermore, for any prescribed nonnegative
integer n, there is a non-finitely generated Puiseux monoid with exactly n atoms [6,
Proposition 5.4]. There exist also non-atomic Puiseux monoids with infinitely many
atoms; see, for instance, [6, Example 3.5]. Moreover, as Theorem 3.5 states, there are
atomic Puiseux monoids whose set of atoms is dense in R≥0 with respect to the standard
subspace topology. This variety of unexpected atomic phenomena makes the family
of Puiseux monoids a vast source of examples that might be useful to test sharpness
of theorems and falsehood of conjectures in the areas of commutative semigroups and
factorization theory.
As addition is a continuous operation with respect to the standard topology of R,
the subspace topology inherited by a Puiseux monoid M is intrinsically connected to
the algebraic structure of M. In particular, if one is willing to study the atomic decomposition of elements of M, a neighborhood of 0 might provide substantial information.
Indeed, if 0 is not a limit point of M, then it is not hard to see that M is atomic.
However, when 0 is a limit point of M, the atomic structure of M might become considerably intricate. In particular, when 0 is a limit point of M, the additivity of M
Date: January 3, 2017.
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F. GOTTI, M. GOTTI, AND H. POLO

implies that M is dense in R≥0 . Thus, M is dense in R≥0 if and only if 0 is a limit
point of M. This observation motivates the following definition.
Definition 1.1. A Puiseux monoid is dense if it has 0 as a limit point.
The atomicity of some families of dense Puiseux monoids has been already studied.
For example, every submonoid of the dense Puiseux monoid h1/p | p is primei is atomic
([8, Theorem 5.5]). On the other hand, if r ∈ Q∩(0, 1), then hr n | n is naturali is either
antimatter or atomic ([8, Theorem 6.2]). In this paper, we focus on the study of three
families of dense Puiseux monoids, paying special attention to the atomic structure of
their members. Before beginning the exploration of these targeted families, we give
evidence of how ubiquitous atoms of a Puiseux monoid could be by finding infinitely
many Puiseux monoids whose sets of atoms are dense in R≥0 .
This paper is structured in the following way. In Section 2, we establish the nomenclature we will be using later. Then, in Section 3, we exhibit infinitely many atomic
Puiseux monoids whose sets of atoms are dense in R≥0 . In Section 4, the family of
primary Puiseux monoids is generalized, and the atomicity of the resulting monoids is
explored. Then, in Section 5, we look into those Puiseux monoids satisfying that the
denominators of all its elements are powers of the same prime; we find necessary and
sufficient conditions for the atomicity of subfamilies of these monoids. In the latter
section, we continue the study of the atomic structure of multiplicatively cyclic Puiseux
monoids that was started in [8]. We prove that every multiplicatively cyclic atomic
Puiseux monoid is an FF-monoid (defined in Section 2). Finally, we show that every
multiplicatively cyclic Puiseux monoid that is atomic is necessarily hereditarily atomic
(see Definition 4.1).
2. Preliminary
In this section, we briefly introduce a few concepts related to our exposition as
an excuse to establish the notation we shall be using throughout this paper. For
background material on commutative semigroups the reader can consult [9] by Grillet.
On the other hand, the monograph [5] of Geroldinger and Halter-Koch offers extensive
information on factorization theory of atomic monoids.
The symbol N (resp., N0 ) denotes the set of positive integers (resp., nonnegative
integers), while P denotes the set of primes. For a real number r, we denote the set
{z ∈ Z | z ≥ r} by Z≥r ; in a similar manner we shall use Q≥r , Q>r , and R≥0 . If
r ∈ Q>0 , then we call the unique a, b ∈ N such that r = a/b and gcd(a, b) = 1 the
numerator and denominator of r and denote them by n(r) and d(r), respectively. For
each subset S of Q>0 , we call the sets n(S) = {n(r) | r ∈ S} and d(S) = {d(r) | r ∈ S}
the numerator set and denominator set of S, respectively.
Every time the term “monoid” is used in this sequel, we tacitly assume that the
monoid in question is commutative and cancellative. Thus, we always use additive

THREE FAMILIES OF DENSE PUISEUX MONOIDS

3

notation on a monoid M; in particular, “ + ” denotes the operation of M, while 0
denotes the identity element. We use the symbol M • to denote the set M \ {0}. If
a, b ∈ M, a divides b in M means that there exists c ∈ M such that b = a + c; in this
case we write a |M b. An element a ∈ M \M × is irreducible or an atom if whenever
a = u + v for some u, v ∈ M, one has that either u ∈ M × or v ∈ M × . The set of atoms
of a monoid is relevant enough to deserve a special notation:
A(M) = {a ∈ M | a is an atom of M}.
The monoid M is reduced if M × = {0}. Every monoid in this exposition is reduced.
As a result, A(M) is always contained in each generating set. We write M = hSi when
M is generated by S, and say that M is finitely generated if |S| < ∞. If M = hA(M)i,
then we say that M is atomic. By contrast, a monoid is said to be antimatter if A(M)
is empty. Antimatter domains, studied by Coykendall et al. in [3], are defined in a
similar fashion.
The free abelian monoid on A(M) is denoted by Z(M) and called factorization
monoid of M; the elements of Z(M) are called factorizations. If z = a1 . . . an ∈ Z(M)
for some n ∈ N0 and a1 , . . . , an ∈ A(M), then n is the length of the factorization z;
the length of z is commonly denoted by |z|. The unique homomorphism
φ : Z(M) → M satisfying φ(a) = a for all a ∈ A(M)
is called the factorization homomorphism of M. For x ∈ M,
Z(x) = φ−1 (x) ⊆ Z(M)
is the set of factorizations of x. If x ∈ M satisfies |Z(x)| = 1, then we say that x has
unique factorization. By definition, we set Z(0) = {0}. Note that the monoid M is
atomic if and only if Z(x) is not empty for all x ∈ M. The monoid M satisfies the
finite factorization property if for all x ∈ M the set Z(x) is finite; in this case we also
say that M is an FF-monoid. The next proposition follows from [5, Theorem 3.1.4].
Proposition 2.1. Every finitely generated reduced monoid is an FF-monoid.
For each x ∈ M, the set of lengths of x is defined by
L(x) = {|z| : z ∈ Z(x)}.
The set of lengths is an arithmetic invariant of atomic monoids that has been very well
studied in recent years (see [1, 2] and the references therein). If L(x) is a finite set for
all x ∈ M, we say that M satisfies the bounded factorization property, in which case,
we call M a BF-monoid. Note that every FF-monoid is also a BF-monoid.
A very special family of atomic monoids is that one comprising all numerical semigroups, cofinite submonoids of the additive monoid N0 . Each numerical semigroup has
a unique minimal set of generators, which is finite. Moreover, if {a1 , . . . , an } is the
minimal generating set for a numerical semigroup N, then it follows that A(N) =
{a1 , . . . , an } and gcd(a1 , . . . , an ) = 1. As a consequence, every numerical semigroup is

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F. GOTTI, M. GOTTI, AND H. POLO

atomic and contains only finitely many atoms. The Frobenius number of N, denoted
by F (N), is the minimum natural n such that Z≥n ⊆ N. Readers can find an excellent
exposition of numerical semigroups in [4] by Garcı́a-Sánchez and Rosales.
As mentioned in the introduction, Puiseux monoids are a natural extension of numerical semigroups. Puiseux monoids are not always atomic; for instance consider
h1/2n | n ∈ Ni. However, a Puiseux monoid is atomic provided 0 is not a limit point.
We say that a Puiseux monoid M is strongly bounded if it can be generated by a set
of positive rationals S such that n(S) is bounded.
For a prime p, the p-adic valuation on Q is the map defined by vp (0) = ∞ and
vp (r) = vp (n(r)) − vp (d(r)) for r 6= 0, where for n ∈ N the value vp (n) is the exponent
of the maximal power of p dividing n. It follows immediately that the p-adic valuation
satisfies that
(2.1)

vp (r1 + · · · + rn ) ≥ min{vp (r1 ), . . . , vp (rn )}

for every n ∈ N and r1 , . . . , rn ∈ Q>0 . We say that a Puiseux monoid M is finite if
there exists a finite subset P of P such that vp (r) ≥ 0 for all p ∈ P \P and r ∈ M • .
There are only a few distinguished strongly bounded finite Puiseux monoids that are
atomic, as the next theorem indicates.
Theorem 2.2. [6, Theorem 5.8] Let M be a strongly bounded finite Puiseux monoid.
Then M is atomic if and only if M is isomorphic to a numerical semigroup.
Example 2.3. For q ∈ P, consider the following Puiseux monoids:




1
⌊p/2⌋
′
M=
p∈P .
n ∈ N and M =
qn
p
It is follows immediately that M is finite, strongly bounded, and antimatter. On the
other hand, it is not hard to verify that M ′ is atomic but it is neither finite nor strongly
bounded.
More information about the family of Puiseux monoids can be found in [6] and [8],
where their atomic structure was studied.

3. Density of Puiseux Monoids
Recall that we defined a dense Puiseux monoid to be a Puiseux monoid having 0 as
a limit point. As mentioned in the introduction, a Puiseux monoid is dense if and only
if it is topologically dense in R≥0 with respect to the inherited standard topology.
Proposition 3.1. An additive submonoid of Q≥0 is a dense Puiseux monoid if and
only if it is topologically dense in R≥0 .

THREE FAMILIES OF DENSE PUISEUX MONOIDS

5

Proof. The forward implication follows immediately. Suppose, conversely, that M is a
Puiseux monoid having 0 as a limit point. Let {rn } be a sequence in M • converging
to 0. Fix any p ∈ R>0 . Let us check that p is a limit point of M. Take ǫ > 0. Because
lim rn = 0, there exists n ∈ N such that rn < min{p, ǫ}. Let m be the maximum
integer such that p − mrn > 0, and take r = mrn . By the maximality of m,
0 < p − r = p − (m + 1)rn + rn ≤ rn < ǫ.
Since for an arbitrary ǫ we have found r ∈ M \ {p} such that |p − r| < ǫ, we gets that
p is a limit point of M. Hence M is a dense Puiseux monoid.

The following corollary follows directly from the fact that every Puiseux monoid that
does not have 0 as a limit point is atomic.
Corollary 3.2. Every non-atomic Puiseux monoid is topologically dense in R≥0 .
The rest of this section is dedicated to establishing the fact that there exist infinitely
many non-isomorphic Puiseux monoids whose sets of atoms is dense in R≥0 . If two
Puiseux monoids M and M ′ are isomorphic, we write M ∼
= M ′ . It follows by the
next lemma, that two Puiseux monoids are isomorphic if and only if they are rational
multiples of each other.
Lemma 3.3. The homomorphisms of Puiseux monoids are precisely those given by
rational multiplication.
Proof. Let M and M ′ be two Puiseux monoids, and let α : M → M ′ be a homomorphism. If α is the trivial homomorphism, then it is multiplication by 0. Therefore let
us assume that α is not the trivial homomorphism, which implies that M 6= {0}. As
N = M ∩ N0 is a nontrivial submonoid of the additive monoid of nonnegative integers, it has a nonempty minimal set of generators, namely {s1 , . . . , sk }. Because α is
not the zero homomorphism, there exists j ∈ {1, . . . , k} such that α(sj ) 6= 0. Take
a = α(sj )/sj . For x ∈ M • and n1 , . . . , nk ∈ N0 satisfying that n(x) = n1 s1 + · · · + nk sk ,
the fact that si α(sj ) = α(si sj ) = sj α(si ) for each i = 1, . . . , k implies that
k

k

1
1 X
α(sj )
1 X
α(x) =
ni α(si ) =
ni si
α(n(x)) =
= xa.
d(x)
d(x) i=1
d(x) i=1
sj
Hence the homomorphism α is multiplication by the rational a. On the other hand, it
follows immediately that, for all rational r, the map M → M ′ defined by x 7→ rx is a
homomorphism as long as rM ⊆ M ′ .

Lemma 3.4. Let P and Q be disjoint infinite sets of primes, and let MP = hap | p ∈ P i
and MQ = hbq | q ∈ Qi be Puiseux monoids such that for all p ∈ P and q ∈ Q, d(ap )
and d(bq ) are powers of p and q, respectively. Then MP ≇ MQ .

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F. GOTTI, M. GOTTI, AND H. POLO

Proof. Suppose, by way of contradiction, that MP ∼
= MQ . By Lemma 3.3, there exists
a rational r such that MP = rMQ . If q is a prime in Q such that q ∤ n(r), then rbq
would be an element of MP such that d(rbq ) is a power of q and, therefore, q ∈ P . But
this contradicts the fact that P ∩ Q is empty.

In order to generate a dense Puiseux monoid it suffices to take a sequence of positive
rationals having 0 as a limit point. Let P = {Pn | n ∈ N} be a collection of infinite
subsets of primes such that Pi ∩ Pj is empty for i 6= j. Now for each j ∈ N, consider
the Puiseux monoid Mj = h1/p | p ∈ Pj i. Because every Pj is infinite, each Mj is
dense. Moreover, Mi ≇ Mj for i 6= j; this is an immediate consequence of Lemma 3.4.
Hence we can conclude that there are countably many non-isomorphic dense Puiseux
monoids. The next theorem will strengthen this observation. Although the existence
of dense Puiseux monoids is an straightforward fact, there still remains the question
as to whether there exists an atomic Puiseux monoid whose set of atoms is dense in
R≥0 . It turns out that there are infinitely many atomic Puiseux monoids satisfying
this stronger density condition. Without further ado, let us present the main result of
this section.
Theorem 3.5. There are infinitely many non-isomorphic Puiseux monoids whose sets
of atoms are dense in R≥0 .
Proof. First, let us check that the set S = {m/pn | m, n ∈ N and p ∤ m} is dense in R≥0
for every p ∈ P. To see this, take ℓ ∈ R>0 and then fix ǫ > 0. Now take n, m ∈ N such
that 1/pn < ǫ and m/pn < ℓ ≤ (m + 1)/pn . It follows immediately that the element
s = m/pn of S satisfies that |ℓ − m/pn | < ǫ. Since ǫ was arbitrarily taken, ℓ is a limit
point of S. It is obvious that 0 is also a limit point of S. Hence S is dense in R≥0 .
Now take {rn } to be a sequence of positive rationals with underlying set R dense
in R≥0 . Also, consider the collection P = {Pn | n ∈ N} of infinite sets of primes such
that Pi ∩ Pj is empty for i 6= j. For each j ∈ N, let Pj = {pjk | k ∈ N}. Now for every
j ∈ N and pjk ∈ Pj the set


m
m, n ∈ N and p ∤ m
pnjk
is dense in R≥0 . Therefore, for every natural k, there exist naturals mk and nk satisfying
that |rk − mk /pnjkk | < 1/k. Consider the Puiseux monoid


mk
(3.1)
Mj =
k∈N .
pnjkk
Because distinct generators in (3.1) have powers of distinct primes in their denominators, it follows that Mj is atomic and A(Mj ) = {mk /pnjkk | k ∈ N}. Finally, we are led
to verify that A(Mj ) is dense in R≥0 . To do this take x ∈ R≥0 and then fix ǫ > 0.

THREE FAMILIES OF DENSE PUISEUX MONOIDS

7

Since R is dense in R≥0 , there exists a natural k large enough such that 1/k < ǫ/2 and
|x − rk | < ǫ/2. Consequently, |rk − mk /pnjkk | < 1/k < ǫ/2, which implies that
x−

mk
mk
nk < |x − rk | + rk − nk < ǫ.
pjk
pjk

This means that A(Mj ) is dense in R≥0 , as desired. By Lemma 3.4 the Puiseux monoids
in P are pairwise non-isomorphic. Therefore there exist infinitely many atomic Puiseux
monoids whose sets of atoms are dense in R≥0 .

4. k-primary Puiseux Monoids
There are Puiseux monoids satisfying that each of their submonoids are atomic.
These Puiseux monoids are relevant enough to deserve a name.
Definition 4.1. A Puiseux monoid M is hereditarily atomic if every submonoid of M
is atomic.
We often show that a monoid is atomic by finding a hereditarily atomic monoid
containing it (see, e.g., the proof of Proposition 6.8). It follows immediately that if a
Puiseux monoid is not dense, then it is hereditarily atomic. In addition, Proposition 4.4
and Proposition 6.8 provide families of dense hereditarily atomic Puiseux monoids.
Note that every hereditarily atomic Puiseux monoid is, in particular, atomic. However,
not every atomic Puiseux monoid is hereditarily atomic, as we shall see momentarily.
Example 4.2. Let {pn } be a sequence listing the odd prime numbers. Then consider
the Puiseux monoid


1
M = hAi, where A =
n∈N .
2 n pn
Since each odd prime divides exactly one element of the set d(A), it follows that
A(M) = A. Thus, M is atomic. On the other hand, the element 1/2n is the sum of
pn copies of the atom 1/(2n pn ) for every n ∈ N and, therefore, the antimatter monoid
h1/2n | n ∈ Ni is a submonoid of M. Hence M fails to be hereditarily atomic.
Definition 4.3. Let P be a nonempty set of primes. A Puiseux monoid M is primary
over P if there exists a set of positive rationals S such that M = hSi and d(S) = P .
Within the scope of this paper, the term primary monoid refers to a Puiseux monoid
that is primary over some nonempty set of primes. Let us remark that if a Puiseux
monoid M is primary, then there exists a unique P ⊆ P such that M is primary over P .
We call the monoid h1/p | p is primei the elementary primary Puiseux monoid. It was
proved in [8, Section 5] that the elementary primary Puiseux monoid is hereditarily
atomic. The next proposition is an immediate consequence of this fact.

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F. GOTTI, M. GOTTI, AND H. POLO

Proposition 4.4. Every primary monoid is hereditarily atomic.
Let us present a natural way of generalizing the concept of a primary Puiseux monoid,
and then provide some insight on the atomicity of this generalized family.
Definition 4.5. For k ∈ N, a Puiseux monoid M is a k-primary monoid if it can be
generated by a set S such that d(s) is the product of k distinct primes for all s ∈ S.
Observe that the 1-primary monoids are precisely the primary monoids. For a set
S and a nonnegative integer n, let us denote by Sn the collection of subsets of S of
cardinality n. The elementary k-primary monoid is the monoid

 
1
P
Mk = hSk i, where Sk =
.
{p1 , . . . , pk } ∈
k
p1 · · · pk
We call the elements of Sk elementary generators of Mk . Although Proposition 4.4
might suggest that k-primary monoids are hereditarily atomic, this is far from being
the case.
Proposition 4.6. For every k ≥ 2, the elementary k-primary monoid is antimatter.
Proof. First, we show that for every natural N and distinct primes p and q there exist
m, n ∈ N and p′ , q ′ ∈ P with q ′ > p′ > N such that
(4.1)

p′ q ′ = mqq ′ + npp′ + pq.

Let p and q be such two distinct primes. Since gcd(p, q) = 1, Dirichlet’s theorem
on arithmetic progressions of primes ensures the existence of a natural m such that
p′ = mq + p is a prime greater than N. Dirichlet’s theorem comes into play again to
yield a natural n such that q ′ = np′ + q is a prime. Therefore one finds that
p′ q ′ = mqq ′ + pq ′ = mqq ′ + p(np′ + q) = mqq ′ + npp′ + pq.
Now consider the elementary 2-primary monoid M2 . If p and q are distinct primes,
then by the argument given above, there exist m, n ∈ N and p′ , q ′ ∈ P satisfying
q ′ > p′ > max{p, q} such that the identity (4.1) holds. Dividing both sides of (4.1) by
pqp′ q ′ , we obtain
1
1
1
1
= m ′ + n ′ + ′ ′.
pq
pp
qq
pq
As a consequence, no element of M2 can be an atom, which means that M2 is an
antimatter Puiseux monoid.
At this point we are in a position to check the more general fact that, for each
k ≥ 2, the elementary k-primary monoid Mk is antimatter. To do this, fix an arbitrary
elementary generator (p1 · · · pk )−1 of Mk . As before, there exist m, n ∈ N and p′ , q ′ ∈ P
satisfying that q ′ > p′ > max{p1 , . . . , pk } and
1
1
1
1
(4.2)
= m ′ + n ′ + ′ ′.
p1 p2
p1 p
p2 q
pq

THREE FAMILIES OF DENSE PUISEUX MONOIDS

Multiplying both sides of (4.2) by (p3 · · · pk )−1 we get
1
1
1
1
=m ′
+n ′
+ ′ ′
.
p1 p2 · · · pk
p1 p p3 · · · pk
p2 q p3 · · · pk p q p3 · · · pk
Hence no element of Mk can be an atom and, thus, Mk is antimatter.

9



The next example sheds some light upon the fact that k-primary monoids generated
by infinitely many elementary generators of the elementary k-primary monoid may not
be antimatter.
Example 4.7. Let {pn } be an enumeration of P, and let k be a positive integer.
Consider the k-primary monoid
Y

k
1
M = hAi, where A =
n ∈ N0 .
p
nk+i
i=1
Because each prime divides exactly one element of the set d(A), it follows that every
element of A is an atom of M. Therefore M is atomic.
As Example 4.8 illustrates, a k-primary monoid generated by multiples of the elementary generators of Mk might be atomic.
Example 4.8. Let {pn } be an enumeration of P. For k ∈ N, consider the following
k-primary monoid:

 
X 1
N
.
M = hAi, where A = aS =
S∈
ps
k
s∈S
It is not hard to see that d(A) is the set of all possible products of k distinct primes. On
the other hand, if aS = c1 aS1 + · · · + cn aSn for some n, c1 , . . . , cn ∈ N and S1 , . . . , Sn ∈
N
, then for every s ∈ S there exists j ∈ {1, . . . , n} such that ps | d(aSj ). Therefore
k
(4.3)

aS =

n
X
X 1
X 1
ci
≤
.
p
p
s
s
i=1
s∈S
s∈S
i

Note that equality in (4.3) holds if and only if n = 1, c1 = 1, and S1 = S. Hence
A(M) = A, and M is atomic.
5. p-adic Puiseux Monoids
The simplest representatives of dense Puiseux monoids that are not isomorphic to numerical semigroups are of the form M = h1/pn | n ∈ Ni, where p is a prime. Although
these representatives happen to be antimatter, they contain plenty of submonoids with
a very diverse atomic structure. In this section we delve into the atomicity of submonoids of M.

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F. GOTTI, M. GOTTI, AND H. POLO

Definition 5.1. Let p be a prime. We say that a Puiseux monoid M is p-adic if d(x)
is a power of p for all x ∈ M • .
We use the term p-adic monoid as a short for p-adic Puiseux monoid. Throughout
this section, every time that we define a p-adic monoid by specifying a sequence of
generators {rn }, we shall implicitly assume that {d(rn )} increases to infinity; this
assumption comes without loss of generality because in order to generate a Puiseux
monoid we only need to repeat each denominator finitely many times. On the other
hand, lim d(rn ) = ∞ does not affect the generality of the results we prove in this
section for if {d(rn )} is a bounded sequence, then the p-adic monoid generated by {rn }
is finitely generated and, therefore, isomorphic to a numerical semigroup.
Strongly bounded p-adic monoids happen to have only finitely many atoms (cf.
Theorem 2.2), as revealed by the next proposition.
Proposition 5.2. A strongly bounded p-adic monoid has only finitely many atoms.
Proof. For p ∈ P, let M be a strongly bounded p-adic monoid. Let {rn } be a generating
sequence for M with underlying set R satisfying that n(R) = {n1 , . . . , nk } for some
k, n1 , . . . , nk ∈ N. For each i ∈ {1, . . . , k}, take Ri = {rn | n(rn ) = ni } and Mi = hRi i.
The fact that R ⊆ M1 ∪ · · · ∪ Mk , along with A(M) ∩ Mi ⊆ A(Mi ), implies that
A(M) ⊆

k
[

A(Mi ).

i=1

Thus, showing that A(M) is finite amounts to verifying that |A(Mi )| < ∞ for each
i = 1, . . . , k. Fix i ∈ {1, . . . , k}. If Mi is finitely generated, then |A(Mi )| < ∞.
Let us assume, therefore, that Mi is not finitely generated. This means that there
exists a strictly increasing sequence {αn } such that Mi = hni /pαn | n ∈ Ni. Because
ni /pαn = pαn+1 −αn (ni /pαn+1 ), the monoid Mi satisfies that |A(Mi )| = 0. Hence we
conclude that A(M) is finite.

We are now in a position to give a necessary condition for the atomicity of p-adic
monoids.
Theorem 5.3. Let p ∈ P, and let M be an atomic p-adic monoid satisfying that
A(M) = {rn | n ∈ N}. If lim rn = 0, then lim n(rn ) = ∞.
Proof. Set an = n(rn ) and pαn = d(rn ) for every natural n. Suppose, by way of
contradiction, that lim an 6= ∞. Then there exists m ∈ N such that an = m for
infinitely many n ∈ N. For each positive divisor d of m we define the Puiseux monoid


akn
Md = hSd i, where Sd =
akn = m or gcd(m, akn ) = d .
pαkn

THREE FAMILIES OF DENSE PUISEUX MONOIDS

11

Observe that A(M) is included in the union of the Md . On the other hand, the fact
that A(M) ∩ Md ⊆ A(Md ) for every d dividing m implies that
[
(5.1)
A(M) ⊆
A(Md ).
d|m

Because A(M) contains infinitely many atoms, the inclusion (5.1) implies the existence
of a divisor d of m such that |A(Md )| = ∞. Set Nd = d1 Md . Since d divides n(q) for
all q ∈ Md , it follows that Nd is also a p-adic monoid. In addition, the fact that Nd is
isomorphic to Md implies that |A(Nd)| = |A(Md )| = ∞. After setting bn = akn /d and
βn = αkn for every natural n such that either akn = m or gcd(m, akn ) = d, we have


bn
Nd =
n∈N .
pβ n
As an = m for infinitely many n ∈ N, the sequence {βn } is an infinite subsequence of
{αn } and, therefore, it increases to infinity. In addition, as lim an /pαn = 0, it follows
that lim bn /pβn = 0.
Now we argue that A(Nd ) is finite, which will yield the desired contradiction. Take
m′ = m/d. Since there are infinitely many n ∈ N such that bn = m′ , it is guaranteed
that m′ /pn ∈ Nd for every n ∈ N. In addition, gcd(m′ , bn ) = 1 for each bn 6= m′ . If
bn 6= m′ for only finitely many n, then Nd is strongly bounded and Proposition 5.2
ensures that A(Nd ) is finite. Suppose otherwise that gcd(bn , m′ ) = 1 (i.e., bn 6= m′ )
for infinitely many n ∈ N. For a fixed i with bi 6= m′ take j ∈ N satisfying that
gcd(bj , m′ ) = 1 and large enough so that bi pβj −βi > bj m′ ; the existence of such an index
j is guaranteed by the fact that lim bn /pβn = 0. As bi pβj −βi > bj m′ > F (hbj , m′ i), there
exist positive integers x and y such that bi pβj −βi = xbj + ym′ , that is
bj
m′
bi
=
x
+
y
.
pβ i
pβ j
pβ j
As bj /pβj , m′ /pβj ∈ Nd• , it follows that bi /pβi ∈
/ A(Nd ). Because i was arbitrarily taken,
Nd is antimatter. In particular, A(Nd ) is finite, which leads to a contradiction.

The conditions lim rn = 0 and lim n(rn ) = ∞ are not enough to guarantee that the
non-finitely generated p-adic monoid M is atomic. The next example sheds some light
upon this observation.
Example 5.4. For an odd prime p, consider the p-adic monoid
 2n

n
p − 1 p2 + 1
(5.2)
M=
, 2n+1
n∈N .
p2n+1
p
n

n

Observe that the sequence of numerators {p2 − 1, p2 + 1} increases to infinity while
the sequence of generators of M converges to zero. Also, notice that for every n ∈ N,
n
n
p2 − 1 p2 + 1
2
= 2n+1 + 2n+1 ∈ M.
p2n
p
p

12

F. GOTTI, M. GOTTI, AND H. POLO

Now we can see that M is not atomic; indeed, M is antimatter, which immediately
follows from the fact that
n
n
p2 ± 1
p2 ± 1 2
=
.
2 p2n+1
p2n+1
The next proposition yields a necessary and a sufficient condition for the atomicity
of p-adic monoids having generating sets whose numerators are powers of the same
prime.
Proposition 5.5. Let p and q be two different primes, and let M = hrn | n ∈ Ni be a
p-adic monoid such that n(rn ) is a power of q for every n ∈ N. Then
(1) if M is atomic, then lim n(rn ) = ∞;
(2) if lim n(rn ) = ∞ and {rn } is decreasing, then M is atomic.
Proof. Define the sequences {αn } and {βn } such that pαn = d(rn ) and q βn = n(rn ). To
check condition (1), suppose, by way of contradiction, that lim n(rn ) 6= ∞. Therefore
there is a natural j such that n(rn ) = q j for infinitely many n ∈ N. This implies that
q j /pn ∈ M for every n ∈ N. Thus, for every x ∈ M • such that n(x) = q m ≥ q j , one
can write
qm
qj
x=
= pq m−j
∈
/ A(M).
d(x)
pd(x)
As a result, every a ∈ A(M) satisfies that n(a) < q j . This immediately implies that
A(M) is finite. As M is atomic with |A(M)| < ∞, it must be finitely generated, which
is a contradiction.
Let us verify condition (2). Consider the subsequence {kn } of naturals satisfying
that n(rkn ) < n(ri ) for every i > kn . It follows immediately that the sequence {n(rkn )}
/ {kn }. Because
is increasing. We claim that M = hrkn | n ∈ Ni. Suppose that j ∈
lim n(rn ) = ∞ there are only finitely many indices i ∈ N such that n(ri ) ≤ n(rj ),
and it is easy to see that the maximum of such indices, say m, belongs to {kn }. As
ri = pαm −αi q βi −βm rm , it follows that ri ∈ hrkn | n ∈ Ni. Hence M = hrkn | n ∈ Ni.
Therefore it suffices to show that rkn ∈ A(M) for every n ∈ N. If
t

(5.3)

X q β ki
q β kn
=
ci αk ,
pαkn
p i
i=1

for some t, c1 , . . . , ct ∈ N0 , then t ≥ n, c1 = · · · = cn−1 = 0, and cn ∈ {0, 1}. If cn = 0,
then by applying the q-adic valuation map to both sides of (5.3) we immediately
obtain a contradiction. Thus, cn = 1, which implies that rkn is an atom. Hence M is
atomic.


THREE FAMILIES OF DENSE PUISEUX MONOIDS

13

6. Multiplicatively Cyclic Puiseux Monoids
The study of the atomic structure of multiplicatively cyclic Puiseux monoids was
initiated in [8]. Our purpose in this section is to continue the exploration of this family
of Puiseux monoids. We shall prove that atomic members of this family are FF-monoid
and, therefore, hereditarily atomic. We also introduce a natural generalization of this
family and explore the atomic structure of its members.
Definition 6.1. For r ∈ Q>0 , the multiplicatively r-cyclic monoid is the Puiseux
monoid generated by the positive powers of r.
A multiplicatively cyclic monoid is a Puiseux monoid that is multiplicatively r-cyclic
for some r ∈ Q>0 . The multiplicatively r-cyclic monoid is antimatter when n(r) = 1;
otherwise it is atomic [8, Theorem 6.2].
Given a monoid M and x ∈ M, we define the set of atoms of x to be
AM (x) = {a ∈ A(M) : a |M x}.
When there is no risk of ambiguity, we write A(x) instead of AM (x). The set of atoms
of an element can be seen as a local statistic that is naturally related to Z(x).
Lemma 6.2. Let M be an atomic reduced monoid and x ∈ M. Then |A(x)| < ∞ if
and only if |Z(x)| < ∞.
Proof. To show the forward implication, take x ∈ M such that AM (x) is a finite set,
say AM (x) = {a1 , . . . , an }. Consider the submonoid N = ha1 , . . . , an i of M. Since
A(M) ∩ N ⊆ A(N) it follows that A(N) = {a1 , . . . , an }. By Proposition 2.1, the
monoid N is an FF-monoid. Now the fact that AM (x) = AN (x) implies that |ZM (x)| =
|ZN (x)| < ∞. For the reverse implication, note that an atom of A(x) must show up in
at least one factorization in Z(x). Hence if Z(x) is finite so is A(x).

Theorem 6.3. Every multiplicatively cyclic atomic monoid is an FF-monoid.
Proof. Take r ∈ Q>0 such that the multiplicatively r-cyclic monoid Mr is atomic. By
Lemma 6.2, proving that Mr is an FF-monoid amounts to showing that A(x) is finite
for all x ∈ Mr . Set a = n(r) and b = d(r). Since Mr is atomic either it is cyclic or
min{a, b} ≥ 2. Arguing that Mr is an FF-monoid when either Mr is cyclic or 2 ≤ b < a
is straightforward. Therefore let us assume that 2 ≤ a < b.
Suppose, by way of contradiction, that there exists x ∈ Mr such that A(x) contains
infinitely many atoms. Let
a t1
a tn
c1
∈ Z(x),
+ · · · + cn
b
b
where ci , ti ∈ N for each i = 1, . . . , n. After simplifying if necessary, we can assume that
t1 < · · · < tn and a ∤ ci for any i ∈ {1, . . . , n}. Because Mr is atomic and |A(x)| = ∞,
there exists
a lm
a l1
∈ Z(x)
+ · · · + dm
d1
b
b

14

F. GOTTI, M. GOTTI, AND H. POLO

such that lm > tn , where dj , tj ∈ N for each j = 1, . . . , m. As before we can perform
necessary simplifications to have l1 < · · · < lm and a ∤ dj for any j ∈ {1, . . . , m};
observe that such simplifications do not affect the fact that lm > tn . Therefore we have
(6.1)

n
m
X
X
ati
alj
ci t =
dj lj .
bi
b
i=1
j=1

From now on, we will not use the fact that tn < lm , but only that tn 6= lm . By canceling
terms in (6.1) if necessary, we can assume that t1 6= l1 , say t1 < l1 . Since a ∤ c1 , there
exist a prime p and a natural α such that pα | a, pα+1 ∤ a, and pα ∤ c1 . The fact that
t1 < ti for each i ∈ {2, . . . , n} implies vp (c1 at1 /bt1 ) = vp (x). Taking p-adic valuation in
(6.1), one finds that
 t1 
  lj 

a
a
αt1 + (α − 1) ≥ vp c1 t1 = vp (x) ≥ min vp dj lj
j ∈ {1, . . . , m} ≥ α(t1 + 1),
b
b
which is a contradiction. Hence |A(x)| < ∞ for all x ∈ Mr . Lemma 6.2 ensures now
that Mr is, indeed, an FF-monoid.

Corollary 6.4. Every atomic multiplicatively cyclic Puiseux monoid is a BF-monoid.
To make sure that Corollary 6.4 is not just a restatement of Theorem 6.3, let us
exhibit an example of a Puiseux monoid that is a BF-monoid but fails to be an FFmonoid.
Example 6.5. Let P be the set of odd primes, and consider the primary monoid


⌊p/2⌋ p − ⌊p/2⌋
,
p∈P .
M = hAi, where A =
p
p
As M is a primary monoid it is atomic. In addition, it is not hard to verify that
A(M) = A. As
⌊p/2⌋ p − ⌊p/2⌋
1=
+
p
p
for each p ∈ P , we have that |A(1)| = ∞. Also, notice that a ≥ 1/3 for every a ∈ A(M).
This implies that no element x ∈ M can be the sum of more than ⌊3x⌋ atoms, i.e.,
L(x) ⊆ {1, . . . , ⌊3x⌋}. Because |L(x)| < ⌊3x⌋ for all x ∈ M the Puiseux monoid M is
a BF-monoid that fails to be an FF-monoid.
In [7, Proposition 6.4] it was proved that every positive BF-monoid of an ordered
field is hereditarily atomic. As a Puiseux monoid is a positive monoid of the ordered
field Q, we obtain the following result.
Proposition 6.6. If a Puiseux monoid is a BF-monoid, then it is hereditarily atomic.

THREE FAMILIES OF DENSE PUISEUX MONOIDS

15

The atomicity of multiplicatively cyclic monoids was described in [8]. If a multiplicatively r-cyclic monoid Mr is not dense (i.e., r ≥ 1), it follows immediately that
Mr is hereditarily atomic. We can extend now this result to dense Puiseux monoid as
a direct consequence of Corollary 6.4 and Proposition 6.6.
Corollary 6.7. Every atomic multiplicatively cyclic monoid is hereditarily atomic.
The fact that atomic multiplicatively cyclic monoids are hereditarily atomic can be
further extended to a more general family of Puiseux monoids.
Proposition 6.8. Let k ∈ N and r1 , . . . , rk ∈ Q>0 such that gcd(n(r1 ), . . . , n(rk )) 6= 1.
Then the Puiseux monoid hr1n , . . . , rkn | n ∈ Ni is hereditarily atomic.
Proof. Let us denote hr1n , . . . , rkn | n ∈ Ni by M. Also, take p to be a prime number
dividing gcd(n(r1 ), . . . , n(rk )). We will verify that the Puiseux monoid
n


p
′
n∈N
M =
d(r1 ) . . . d(rk )
contains M. For each i = 1, . . . , k there exists qi ∈ N such that n(ri ) = pqi . Therefore
it follows that

m Qk
m

m
pq
p
i
j=1 d(rj )
m
ri =
qi
=
∈ P.
d(ri )
d(ri )m
d(r1 ) . . . d(rk )
for every natural m. This implies that M is a submonoid of M ′ . By Corollary 6.7, the
Puiseux monoid M ′ is hereditarily atomic. Since M is a submonoid of M ′ , it must be
also hereditarily atomic.

Finally, note that the condition on the numerators in Proposition 6.8 is not superfluous as the following example reveals.
Example 6.9. Consider the Puiseux monoid
n 
m

2
3
M=
,
7 · 11
7 · 11


n, m ∈ N .

Fix k ∈ N. For such k we can bound the Frobenius number F (N) of the numerical
semigroup N = h2k , 3k i in the following way,
F (h2k , 3k i) < (2k − 1)(3k − 1) < 11k .
Therefore there exist α, β ∈ N0 such that α2k + β3k = 11k . This implies that
k
k


α2k + β3k
3
1
2
+β
∈ M.
=
=α
7k
(7 · 11)k
7 · 11
7 · 11
As 1/7k ∈ M for every k ∈ N, the antimatter Puiseux monoid h1/7k | k ∈ Ni is a
submonoid of M. Hence M is not hereditarily atomic.

16

F. GOTTI, M. GOTTI, AND H. POLO

7. Acknowledgments
While working on this paper, the first author was supported by the UC Berkeley Chancellor Fellowship and the second author was under the University of Florida
Mathematics Department Fellowship.
References
[1] J. Amos, S. T. Chapman, N. Hine, and J. Paixao: Sets of Lengths Do not Characterize Numerical
Monoids, Integers 7 (2007), A50.
[2] S. T. Chapman, P. A. Garcı́a-Sánchez, D. Llena, and J. Marshall: Elements in a Numerical
Semigroup with Factorizations with the Same Length, Canadian Math. Bull. 54 (2011), 39–43.
[3] J. Coykendall, D. E. Dobbs, and B. Mullins: On Integral Domains With No Atoms, Communications in Algebra 27 (1999), 5813–5831.
[4] P. Garcı́a-Sánchez and J. Rosales: Numerical Semigroups, Developments in Mathematics, 20,
Springer-Verlag, New York, 2009.
[5] A. Geroldinger and F. Halter-Koch: Non-Unique Factorizations: Algebraic, Combinatorial and
Analytic Theory, Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, Boca Raton,
2006.
[6] F. Gotti: On the Atomic Structure of Puiseux Monoids, Journal of Algebra and Its Applications
16 (2016). [arXiv:1607.01731]
[7] F. Gotti: Increasing Positive Monoids of Ordered Fields Are FF-monoids. [arXiv:1610.08781]
[8] F. Gotti and M. Gotti: Atomicity and Boundedness of Monotone Puiseux Monoids.
[arXiv:1608.04044]
[9] P. A. Grillet: Commutative Semigroups, Advances in Mathematics, vol. 2, Kluwer Academic
Publishers, Boston, 2001.
Mathematics Department, UC Berkeley, Berkeley, CA 94720
E-mail address: felixgotti@berkeley.edu
Mathematics Department, University of Florida, Gainesville, FL 32611
E-mail address: marlycormar@ufl.edu
Mathematics Department, UC Berkeley, Berkeley, CA 94720
E-mail address: haroldpolo@berkeley.edu

