Atomicity and Density of Puiseux Monoids
aa r X i v : . [ m a t h . A C ] A p r ATOMICITY AND DENSITY OF PUISEUX MONOIDS
MARIA BRAS-AMOROS AND MARLY GOTTI
Abstract.
A Puiseux monoid is a submonoid of ( Q , +) consisting of nonnegativerational numbers. Although the operation of addition is continuous with respect tothe standard topology, the set of irreducibles of a Puiseux monoid is, in general,difficult to describe. In this paper, we use topological density to understand howmuch a Puiseux monoid, as well as its set of irreducibles, spread through R ≥ . First,we separate Puiseux monoids according to their density in R ≥ , and we characterizemonoids in each of these classes in terms of generating sets and sets of irreducibles.Then we study the density of the difference group, the root closure, and the conduc-tor semigroup of a Puiseux monoid. Finally, we prove that every Puiseux monoidgenerated by a strictly increasing sequence of rationals is nowhere dense in R ≥ andhas empty conductor. Introduction
A Puiseux monoid is an additive submonoid of ( Q ≥ , +). The first significant appear-ance of Puiseux monoids in commutative algebra seems to date back to the 1970s, whenA. Grams used them in [20] to disprove P. Cohn’s conjecture (see [7]) that every atomicintegral domain satisfies the ACCP (i.e., every ascending chain of principal ideals even-tually stabilizes). However, it was not until recently that Puiseux monoids became thefocus of significant attention in factorization theory because of their rich and complexatomic structure. The first systematic study of Puiseux monoids appeared in [16], andsince then they have been present in the semigroup and factorization theory literature(see, for instance, [6] and [18]). Recent applications of Puiseux monoids to numericalsemigroups and commutative algebra can be found in [15] and [9], respectively.In general, the atomic structure of a Puiseux monoid can be significantly complex.Puiseux monoids range from antimatter monoids (i.e., monoids without atoms) suchas h / n | n ∈ N i to atomic monoids whose sets of atoms are dense in R ≥ (seeExample 3.6). Even though sufficient conditions for atomicity have been found (see [19,Theorem 5.5] and [17, Proposition 4.5]), there is no characterization of atomic Puiseuxmonoids in terms of their generating sets. In this paper we study how much Puiseuxmonoids and, in particular, their sets of atoms, can spread through R ≥ , hoping ourstudy can contribute towards the understanding of their atomic structure. Date : April 23, 2020.2010
Mathematics Subject Classification.
Primary: 20M13; Secondary: 06F05, 20M14, 11B05.
Key words and phrases.
Puiseux monoids, factorization theory, factorization invariants, system ofsets of lengths, set of distances, realization theorem, catenary degree. MARIA BRAS-AMOROS AND MARLY GOTTI
In Section 3, we subclassify Puiseux monoids according to how much they spreadthroughout R ≥ : Puiseux monoids that are dense (in R ≥ ), Puiseux monoids thatare eventually dense (i.e., dense in [ r, ∞ ) for some r > R ≥ ). We characterize members of each of these four classes interms of their sets of atoms and generating sets. In this section we also exhibit anatomic Puiseux monoid whose set of atoms is dense in R ≥ .In the first part of Section 4, we argue that the difference group (resp., the rootclosure) of a Puiseux monoid M is dense in R (resp., in R ≥ ) provided that M isnot finitely generated. Then we fully describe the density of Puiseux monoids withnonempty conductor: such Puiseux monoids are nowhere dense if and only if theyare finitely generated. In the second part of Section 4, we restrict our attention toincreasing Puiseux monoids. A submonoid of ( R ≥ , +) is called increasing providedthat it can be generated by an increasing sequence. Increasing monoids were firststudied in [19] in the context of Puiseux monoids, and then they were investigatedin [3, 4]. Increasing monoids are always atomic [17, Proposition 4.5]. We conclude thispaper proving that every non-finitely generated increasing Puiseux monoid is nowheredense and has empty conductor. 2. Preliminary
General Notation.
In this section, we review most of the notation and termi-nology we shall be using later. The interested reader can consult [21] for backgroundmaterial on commutative semigroups and [13] for extensive information on factoriza-tion theory of atomic monoids. The symbol N (resp., N ) denotes the set of positiveintegers (resp., nonnegative integers), while P denotes the set of primes. For r ∈ R and S ⊆ R , we let S ≥ r denote the set { s ∈ S | s ≥ r } and, in a similar manner, we shalluse the notation S >r . If q ∈ Q > , then we call the unique a, b ∈ N such that q = a/b and gcd( a, b ) = 1 the numerator and denominator of q and denote them by n ( q ) and d ( q ), respectively. For each subset S of Q > , we call the sets n ( S ) = { n ( q ) | q ∈ S } and d ( S ) = { d ( q ) | q ∈ S } the numerator set and denominator set of S , respectively.2.2. Monoids.
Every time the term “monoid” is mentioned here, we tacitly assumethat the monoid in question is commutative and cancellative. Unless we specify other-wise, we use additive notation on any monoid. For a monoid M , we let M • denote theset M \ { } , and we let M × denote the set of invertible elements of M . The monoid M is called reduced when M × = { } . For x, y ∈ M , we say that x divides y in M and write x | M y provided that there exists x ′ ∈ M satisfying y = x + x ′ . An element a ∈ M \ M × is irreducible or an atom if whenever a = u + v for u, v ∈ M , either u ∈ M × or v ∈ M × . The set of atoms of M is denoted by A ( M ). TOMICITY AND DENSITY OF PUISEUX MONOIDS M is a reduced monoid. Let S bea subset of M . If no proper submonoid of M contains S , then S is called a set ofgenerators (or generating set ) of M , in which case we write M = h S i . The monoid M is called finitely generated if M can be generated by a finite set; otherwise, M iscalled non-finitely generated . It is not hard to see that A ( M ) is contained in any setof generators of M . If M = hA ( M ) i , then M is called atomic . By contrast, M iscalled antimatter if A ( M ) is empty. The notion of being antimatter was introducedand studied in [8] in the setting of integral domains.2.3. Factorization Theory.
The (multiplicative) free commutative monoid on A ( M )is denoted by Z ( M ) and called factorization monoid of M ; the elements of Z ( M ) arecalled factorizations . If z = a · · · a ℓ ∈ Z ( M ) for some ℓ ∈ N and a , . . . , a ℓ ∈ A ( M ),then | z | := ℓ is called the length of the factorization z . The unique homomorphism π M : Z ( M ) → M satisfying that π M ( a ) = a for all a ∈ A ( M ) is called the factorizationhomomorphism of M . For each x ∈ M , the set Z ( x ) := π − M ( x ) ⊆ Z ( M )is called the set of factorizations of x . The monoid M is said to be an FF-monoid (or a finite factorization monoid ) if Z ( x ) is finite for all x ∈ M . It follows from [13,Proposition 2.7.8(4)] that every finitely generated 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 wellstudied in recent years (see [11] and the references therein). If L ( x ) is a finite set for all x ∈ M , then M is called a BF-monoid (or a bounded factorization monoid ). Clearly,every FF-monoid is a BF-monoid.2.4.
Numerical and Puiseux Monoids.
A special class of atomic monoids is thatone comprising all numerical monoids , i.e., cofinite submonoids of ( N , +). Each nu-merical monoid has a unique minimal set of generators, which is finite. Let N bea numerical monoid. If { a , . . . , a n } is the minimal set of generators of N , then A ( N ) = { a , . . . , a n } and gcd( a , . . . , a n ) = 1. Thus, every numerical monoid is atomicand contains only finitely many atoms. The Frobenius number of N , denoted by f ( N ),is the minimum n ∈ N such that Z >n ⊆ N . Readers can find an excellent expositionof numerical monoids in [10] and some of their applications in [1].A Puiseux monoid is a submonoid of ( Q ≥ , +). Clearly, every numerical monoid is aPuiseux monoid. In addition, a Puiseux monoid is isomorphic to a numerical monoidif and only if the former is finitely generated [16, Proposition 3.2]. Puiseux monoidsare not always atomic; for instance, consider h / n | n ∈ N i . However, if M is aPuiseux monoid such that 0 is not a limit point of M • , then M is a BF-monoid [17, MARIA BRAS-AMOROS AND MARLY GOTTI
Proposition 4.5] and, therefore, atomic. The atomic structure of Puiseux monoids wasfirst studied in [16] and [19].3.
Atomicity and Density
Our goal is to understand how much the set of atoms of an atomic Puiseux monoidcan spread through R ≥ . To do this it will be convenient to sub-classify Puiseuxmonoids into classes according to their topological density in positive rays of the realline.Let ( X, T ) be a topological space. If Y ⊆ X , then Y naturally becomes a topologicalspace with the subspace topology, in which case we write ( Y, T | Y ). Let A, B ⊆ X suchthat A ⊆ B . Recall that A is a dense set of ( X, T ) (or dense in X ) if the closure of A is X . We say that A is dense in B if A ∩ B is a dense set of ( B, T | B ). Also recall that A is a nowhere dense of ( X, T ) (or nowhere dense in X ) if the interior of its closureis empty. We say that A is nowhere dense in B if A ∩ B is a nowhere dense set of( B, T | B ). Here we only consider the real line R with the Euclidean topology. Definition 3.1.
Let M be a Puiseux monoid. We say that M is(1) dense if M is dense in R ≥ ;(2) eventually dense if there exists r ∈ R ≥ such that M is dense in R >r ;(3) somewhere dense if there exists a nonempty open interval ( r, s ) ⊆ R ≥ suchthat M is dense in ( r, s );(4) nowhere dense if for all r, s ∈ R ≥ with r < s , the set M is not dense in theopen interval ( r, s ).Observe that, a priori , none of the definitions above provides information about theset of atoms or any generating set of M . However, the density of a Puiseux monoidaccording to the previous definitions can be characterized in terms of the topologicaldistribution of its set of atoms. Proposition 3.2.
For an atomic Puiseux monoid M the following conditions are equiv-alent. (1) M is dense. (2) 0 is a limit point of M • . (3) 0 is a limit point of any generating set of M . (4) 0 is a limit point of A ( M ) .Proof. Clearly, (1) implies (2). Since M is reduced, any generating set of M mustcontain A ( M ) and, therefore, (2), (3), and (4) are equivalent. To prove that any ofthe conditions (2), (3), and (4) implies (1), assume that 0 is a limit point of M • .Let { r n } be a sequence in M • converging to 0. Fix p ∈ R > . To check that p is alimit point of M , fix ǫ >
0. Because lim n →∞ r n = 0, there exists n ∈ N such that TOMICITY AND DENSITY OF PUISEUX MONOIDS r n < min { p, ǫ } . Take m = max { k ∈ Z | p − kr n > } , and set r = mr n . Then wehave that 0 < p − r = p − ( m + 1) r n + r n ≤ r n < ǫ . As for any ǫ > r ∈ M \ { p } with | p − r | < ǫ , it follows that p is a limit point of M . So M is a densePuiseux monoid. (cid:3) Corollary 3.3.
If a Puiseux monoid is not dense, then it is atomic.Proof.
It follows immediately from [17, Proposition 4.5], which implies that a Puiseuxmonoid M is a BF-monoid when 0 is not a limit point of M • . (cid:3) Observe that the conditions (1), (2), and (3) in Proposition 3.2 are equivalent evenwhen M is not atomic. In addition, condition (4) always implies all the three previousconditions. However, the atomicity of M is required to obtain condition (4) fromany of the previous conditions; for example, consider the antimatter Puiseux monoid h / n | n ∈ N i .Let us characterize the atomic Puiseux monoids that are somewhere dense. Proposition 3.4.
Let M be an atomic Puiseux monoid. Then the following conditionsare equivalent. (1) A ( M ) is somewhere dense in R ≥ . (2) Each generating set of M is somewhere dense in R ≥ .If any of the above conditions holds, then M is a somewhere dense Puiseux monoid.Proof. Since M is reduced, every generating set contains A ( M ). Therefore (1) im-plies (2). Since M is atomic, A ( M ) is a generating set of M and, therefore, (2)implies (1). The last statement follows straightforwardly. (cid:3) The equivalent conditions in Proposition 3.4 are not superfluous as there are ex-amples of atomic Puiseux monoids whose sets of atoms are not only dense in certaininterval, but they span to a whole interval.
Example 3.5.
Take r, s ∈ R > such that r < s < r . Now consider the Puiseuxmonoid M := h ( r, s ) ∩ Q i . Since 0 is not a limit point of M • , it follows that M isatomic. On the other hand, the condition 2 r > s implies that s is a lower bound for M • + M • . Hence A ( M ) = ( r, s ) ∩ Q .What is even more striking is the existence of an atomic Puiseux monoid whose setof atoms is dense in R ≥ . Example 3.6.
First, we verify that the set S = { m/p n | m, n ∈ N and p ∤ m } is densein R ≥ for every p ∈ P . To see this, take ℓ ∈ R > and then fix ǫ >
0. Now take n, m ∈ N with 1 /p n < ǫ and m/p n < ℓ ≤ ( m + 1) /p n . Clearly, s := m/p n of S satisfiesthat | ℓ − s | < ǫ . Since ǫ was arbitrarily taken, ℓ is a limit point of S . Because 0 is alsoa limit point of S , we conclude that S is dense in R ≥ . MARIA BRAS-AMOROS AND MARLY GOTTI
Now take { r n } to be a sequence of positive rationals with underlying set R dense in R ≥ . Let { p k } be an increasing enumeration of the prime numbers. It follows from theprevious paragraph, that for each k ∈ N , the set (cid:26) mp nk (cid:12)(cid:12)(cid:12)(cid:12) m, n ∈ N and p k ∤ m (cid:27) is dense in R ≥ . Therefore, for every natural k , there exist naturals m k and n k satisfyingthat | r k − m k /p n k k | < /k . Consider the Puiseux monoid(3.1) M := (cid:28) m k p n k k (cid:12)(cid:12)(cid:12)(cid:12) k ∈ N (cid:29) . As distinct generators in (3.1) have powers of distinct primes in their denominators, M must be atomic and A ( M ) = { m k /p n k k | k ∈ N } . To check that A ( M ) is dense in R ≥ , take x ∈ R ≥ and then fix ǫ >
0. Since R is dense in R ≥ , there exists k ∈ N largeenough such that 1 /k < ǫ/ | x − r k | < ǫ/
2. So | r k − m k /p n k k | < /k < ǫ/
2. Then (cid:12)(cid:12)(cid:12)(cid:12) x − m k p n k k (cid:12)(cid:12)(cid:12)(cid:12) < | x − r k | + (cid:12)(cid:12)(cid:12)(cid:12) r k − m k p n k k (cid:12)(cid:12)(cid:12)(cid:12) < ǫ. Hence A ( M ) is dense in R ≥ .We conclude this section proving that being somewhere dense and being eventuallydense are equivalent conditions in the setting of Puiseux monoids. Proposition 3.7.
For a Puiseux monoid M , the following conditions are equivalent. (1) M is eventually dense. (2) M is somewhere dense.Proof. It is clear that (1) implies (2). To verify that (2) implies (1), suppose that M is somewhere dense, i.e., there exists an interval ( r, s ) with r < s such that M istopologically dense in ( r, s ). Since M is closed under addition, it follows that M istopologically dense in ( nr, ns ) for each n ∈ N . Take N ∈ N such that n ( s − r ) > r forevery n ≥ N . In this case, we can see that nr < ( n + 1) r < ns < ( n + 1) s for each n ≥ N which means that R >Nr = ∞ [ n = N (cid:0) nr, ns (cid:1) . Now fix p ∈ R >Nr and ǫ >
0. Take n ∈ N such that p ∈ ( nr, ns ). Since M istopologically dense in ( nr, ns ), there exists x ∈ M ∩ ( nr, ns ) such that | x − p | < ǫ .Hence M is topological dense in R >Nr and, as a result, eventually dense. (cid:3) Corollary 3.8.
Let M be a Puiseux monoid. If A ( M ) is somewhere dense, then M iseventually dense. TOMICITY AND DENSITY OF PUISEUX MONOIDS C is the subset of R one obtains starting withthe interval [0 ,
1] and then removing iteratively the open middle third of each of theintervals in each iteration. Formally, we can write C := [0 , \ ∞ [ n =0 3 n − [ k =0 (cid:18) k + 13 n +1 , k + 23 n +1 (cid:19) . It is well known that the Cantor set is an uncountable nowhere dense subset of [0 , C + C = [0 , E consisting of theend points of all the intervals obtained in each iteration in the construction of C is asubset of C that is dense in C . Example 3.9. As C + C = [0 , C ) + (1 + C ) = [2 ,
4] holds. Letus argue that [(1 + C ) ∩ Q ] + [(1 + C ) ∩ Q ] is dense in the interval [2 , C is just the Cantor set constructed in the interval [1 , E ⊆ Q ∩ C is densein C , we obtain that [(1 + C ) ∩ Q ] + [(1 + C ) ∩ Q ] is dense in [2 , M = h (1 + C ) ∩ Q i . It is clear that A ( M ) = (1 + C ) ∩ Q ,which implies that M is atomic. On the other hand, A ( M ) is nowhere dense as it is asubset of 1 + C . However, we have seen before that M is dense in [2 , Proposition 3.10.
For a Puiseux monoid M , the following statements are equivalent. (1) M is nowhere dense. (2) Each generating set of M is nowhere dense.If any of the above statements holds, then M is atomic and A ( M ) is nowhere dense. Every finitely generated Puiseux monoid is isomorphic to a numerical monoid and,therefore, must be a nowhere dense Puiseux monoids. In addition, there are non-finitelygenerated atomic Puiseux monoids that are nowhere dense. We shall determine a classof such monoids in Theorem 4.9.4.
Difference Group, Closures, and Conductor
Difference Group and the Root Closure.
The difference group of a monoid M , denoted by gp ( M ), is the abelian group (unique up to isomorphism) satisfyingthat any abelian group containing a homomorphic image of M will also contain ahomomorphic image of gp ( M ). When M is a Puiseux monoid, the difference group gp ( M ) can be taken to be a subgroup of ( Q , +), namely, gp ( M ) = { x − y | x, y ∈ M } . Looking at endpoints of the Cantor set was kindly suggested by Jyrko Correa and Harold Polo.
MARIA BRAS-AMOROS AND MARLY GOTTI
The difference group of a Puiseux monoid can be characterized in terms of the de-nominators of its elements. Before stating such a characterization it is convenient tointroduce the notion of the root closure. The root closure f M of a monoid M withdifference group gp ( M ) is defined by f M := (cid:8) x ∈ gp ( M ) | nx ∈ M for some n ∈ N (cid:9) . The monoid M is called root-closed provided that f M = M . The difference group andthe root closure of a Puiseux monoid M were described by Geroldinger et al. in [12]in terms of the set of denominators of M in the following way. Proposition 4.1. [12, Proposition 3.1]
Let M be a Puiseux monoid, and let n =gcd( n ( M • )) . Then (4.1) f M = gp ( M ) ∩ Q ≥ = n (cid:28) d (cid:12)(cid:12)(cid:12)(cid:12) d ∈ d ( M • ) (cid:29) . The normal closure and the complete integral closure are two other algebraic closuresof a monoid that play an important role in semigroup theory and factorization theory.However, we do not formally introduce any of these two notions of closures as, invirtue of [14, Lemma 2.5] and [12, Proposition 3.1], they are both equivalent to theroot closure in the context of Puiseux monoids.
Example 4.2.
Take r ∈ Q > such that d ( r ) ∈ P , and consider the Puiseux monoid M = h r n | n ∈ N i . Since 1 ∈ M , one sees that gcd( n ( M • )) = 1. Also, it is clear that d ( M • ) = { d ( r ) n | n ∈ N } . Then Proposition 4.1 guarantees that f M = (cid:28) d ( r ) n (cid:12)(cid:12)(cid:12)(cid:12) n ∈ N (cid:29) , which is the nonnegative cone of the localization Z d ( r ) of Z at the multiplicative set { q n | n ∈ N } . Observe that M is closed under multiplication, and so it is a cyclicrational semiring. Various factorization invariants of cyclic rational semirings wererecently investigated in [5] by Chapman et al.Although it is difficult in general to determine whether a given Puiseux monoid isdense, a criterion to determine the density of its difference group and its root closurecan be easily established, as the following proposition shows. Proposition 4.3.
Let M be a Puiseux monoid. Then the following statements areequivalent. (1) gp ( M ) is dense in R ; (2) f M is dense in R ≥ ; (3) M is not finitely generated. TOMICITY AND DENSITY OF PUISEUX MONOIDS Proof.
To prove that (1) implies (2), suppose that gp ( M ) is dense in R . It is clearthat every additive subgroup of Q is symmetric with respect to 0. This, togetherwith Proposition 4.1, guarantees that f M = gp ( M ) ∩ Q ≥ is dense in R ≥ , which iscondition (2).Let us argue now that (2) implies (3). Suppose, by way of contradiction, thatthe monoid M is finitely generated. Then d ( M • ) is a finite set, and it follows fromProposition 4.1 that m f M is a submonoid of ( N , +), where m = lcm( d ( M • )). Thereforethe monoid m f M is not dense in R ≥ . However, this clearly implies that f M is not densein R ≥ , which is a contradiction.Finally, we show that (3) implies (1). Assume that M is not finitely generated.Then d ( M • ) contains infinitely many elements as, otherwise, lcm( d ( M • )) M wouldbe a submonoid of ( N , +) and, therefore, finitely generated. Then it follows fromProposition 4.1 that 0 is a limit point of f M • , and it follows from Proposition 3.2 that f M is a dense Puiseux monoid. Hence Proposition 4.1 ensures that gp ( M ) = f M ∪ − f M is dense in R . (cid:3) The Conductor of a Puiseux Monoid.
Let M be a monoid. The conductor of a Puiseux monoid M is defined to be(4.2) c ( M ) := { x ∈ gp ( M ) | x + f M ⊆ M } . It is clear that c ( M ) is a subsemigroup of the group gp ( M ). Although it is moreconvenient for our purposes to define the conductor of a Puiseux monoid in terms ofits root closure, we would like to remark that in general the conductor of a monoid isdefined in terms of its complete integral closure; see for example, [13, Definition 2.3.1].However, recall that the notions of root closure and complete integral closure coincidein the setting of Puiseux monoids. Example 4.4.
Let M be a numerical monoid, and let f ( M ) be the Frobenius numberof M . It follows from Proposition 4.1 that gp ( M ) = Z and f M = N . For n ∈ M with n ≥ f ( M ) + 1, the inclusion n + f M = n + N ⊆ M holds . In addition, for each n ∈ Z with n ≤ f ( M ) the fact that f ( M ) ∈ n + f M implies that n + f M * M . Thus,(4.3) c ( M ) = { n ∈ Z | n ≥ f ( M ) + 1 } . As the equality of sets (4.3) shows, the minimum of c ( M ) is f ( M ) + 1, namely, theconductor number of M as usually defined in the setting of numerical monoids.The conductor of a Puiseux monoid has been recently described in [12] as follows. Proposition 4.5. [12, Proposition 3.2]
Let M be a Puiseux monoid. Then the followingstatements hold. (1) If M is root-closed, then c ( M ) = f M = M . MARIA BRAS-AMOROS AND MARLY GOTTI (2) If M is not root-closed, then set σ = sup f M \ M . (a) If σ = ∞ , then c ( M ) = ∅ . (b) If σ < ∞ , then c ( M ) = M ≥ σ . Puiseux monoids with nonempty conductor have been considered in [12] and, morerecently, in [2]. The density of a Puiseux monoid with nonempty conductor can befully understood with the following criterion.
Proposition 4.6.
Let M be a Puiseux monoid with nonempty conductor. Then M isnowhere dense if and only if M is finitely generated.Proof. For the direct implication, suppose that M is a nowhere dense Puiseux monoid.Since M has nonempty conductor, it follows from Proposition 4.5 that either M isroot-closed or sup f M \ M < ∞ . We consider the following two cases.CASE 1: M is root-closed. Suppose, by way of contradiction, that M is not finitelygenerated. Then Proposition 4.3 guarantees that f M is dense in R ≥ . Since M is root-closed M = f M , and so M is dense in R ≥ . However, this contradicts that M is nowheredense.CASE 2: M is not root-closed. In this case, the inequality sup f M \ M < ∞ must hold.Taking τ := 1 + sup f M \ M , we find that M ≥ τ = f M ≥ τ . Suppose for a contradictionthat M is not finitely generated. Then f M ≥ τ would be dense in R ≥ τ by Proposition 4.3and, therefore, M ≥ τ would also be dense in R ≥ τ . However, this contradicts that M isnowhere dense.The reverse implication follows immediately. Indeed, if M is finitely generated, then d ( M • ) is finite and the set lcm( d ( M • )) M ⊆ N is nowhere dense in R ≥ , which in turnsimplies that M is nowhere dense. (cid:3) Increasing Puiseux Monoids.
A submonoid M of ( R ≥ ) is called increasing if M can be generated by an increasing sequence of real numbers. Clearly, every Puiseuxmonoid generated by an increasing sequence of rationals is an example of an increasingmonoid. Increasing (and decreasing) Puiseux monoids were first studied in [19]. Ifan increasing submonoid of R ≥ (in particular, an increasing Puiseux monoid) can begenerated by an unbounded (resp., bounded) sequence it is called a strongly increasing (resp., weakly increasing ) monoid. Strongly increasing monoids were considered in [3,17] and more recently in [4] under the term “ ω -monoids.” Increasing Puiseux monoidsare always atomic. Indeed, it was proved in [17, Proposition 4.5] that every increasingPuiseux monoid is an FF-monoid.Our goal in this final subsection is to prove that increasing Puiseux monoids arenowhere dense and also that they have empty conductors provided that they are notfinitely generated. First, we will establish some needed lemmas. TOMICITY AND DENSITY OF PUISEUX MONOIDS S be a subset of R , and take α ∈ R . We say that α is a limit point of S fromthe right if for every ǫ ∈ R > the set ( α, α + ǫ ) ∩ S is nonempty. Lemma 4.7.
Let M be an increasing Puiseux monoid. If M has a limit point α fromthe right, then M also has another limit point β from the right with β < α .Proof. Assume that M has a limit point α from the right. Then M cannot be finitelygenerated. Since M is increasing it is an atomic monoid. As M is not finitely generatedthere exists a strictly increasing sequence ( a n ) n ∈ N with underlying set A ( M ). Noticethat if the equality lim n →∞ a n = ∞ held, it would imply that | M ∩ [0 , n ] | < ∞ for every n ∈ N , contradicting that α is a limit point of M from the right. Hence the sequence( a n ) n ∈ N must converge to some ℓ ∈ R > .For each n ∈ N the monoid h a , . . . , a n i is finitely generated and, therefore, theset { s ∈ h a , . . . , a n i | s > α } has a minimum, which we denote by s n . Clearly,the sequence ( s n ) n ∈ N is decreasing. Since α is a limit point of M from the right,lim n →∞ s n = α . Let ( j n ) n ∈ N be the strictly increasing sequence with underlying set { j ∈ N | s j < s j − } . Notice that a j n | M s j n for every n ∈ N . Hence after setting b n := s j n − a j n for every n ∈ N , we obtain that each term of the sequence ( b n ) n ∈ N belongs to M . In addition, b n = s j n − a j n > s j n +1 − a j n +1 = b n +1 for every n ∈ N ,whence ( b n ) n ∈ N is a strictly decreasing sequence. As ( b n ) n ∈ N consists of nonnegativenumbers, lim n →∞ b n = β for some β ∈ R ≥ . So β is a limit point of M from the right.It is clear that β = lim n →∞ b n = lim n →∞ s j n − lim n →∞ a j n = α − ℓ < α , from which thelemma follows. (cid:3) Lemma 4.8.
Let M be an increasing Puiseux monoid. Then M does not contain limitpoints from the right.Proof. Suppose, by way of contradiction, that the set A consisting of all the limit pointsof M from the right is nonempty. Set α = inf A . We will first argue that α is indeedthe minimum of A . To do so take a decreasing sequence ( α n ) n ∈ N whose terms belongto A such that lim n →∞ α n = α . Because each α n belongs to A , for each fixed n ∈ N there exists a strictly decreasing sequence ( b n,j ) j ∈ N whose terms belong to M such thatlim j →∞ b n,j = α n . So for each n ∈ N there exists k n ∈ N such that b n,k n < α n + 1 /n .Therefore ( b n,k n ) n ∈ N is a sequence of elements of M satisfying that lim n →∞ ( b n,k n ) = α .Hence α is a limit point of M from the right and, therefore, α ∈ A . So α = min A .Now Lemma 4.7 guarantees the existence of a limit point β of M from the right suchthat β < α . However, this contradicts the minimality of α . Thus, one can concludethat M has no limit point from the right. (cid:3) We are now ready to prove the main result of this section.
Theorem 4.9.
Let M be an increasing Puiseux monoid. Then the following statementshold. (1) M is nowhere dense. MARIA BRAS-AMOROS AND MARLY GOTTI (2) c ( M ) is nonempty if and only if M is finitely generated.Proof. To argue the statement (1) we proceed by contradiction. Suppose that M is nota nowhere dense Puiseux monoid. Then there exist r, s ∈ R with 0 < r < s such that M is dense in ( r, s ). Take b ∈ M ∩ ( r, s ). Since M is an increasing Puiseux monoid,Lemma 4.8 guarantees the existence of ǫ ∈ R > such that M ∩ [ b, b + ǫ ] = { b } . Thenthe set M ∩ ( b, min { s, b + ǫ } ) is empty. However, this contradicts that M is dense in( r, s ) because the open interval ( b, min { s, b + ǫ } ) is contained in the open interval ( r, s ).Hence M is a nowhere dense Puiseux monoid, as desired.To establish the direct implication of (2), suppose that M be a non-finitely generatedincreasing Puiseux monoid. Then, by Proposition 4.3, the set f M • has 0 as a limitpoint. Since 0 is not a limit point of M • , it follows that M = f M . Hence M cannotbe root-closed. On the other hand, let us argue that f M \ M is unbounded and sosup f M \ M = ∞ . Indeed, let us see that for each r ∈ R there exists an element e c in f M \ M with e c > r . Since the underlying set of M is unbounded, there exists c ∈ M such that c > r . It follows now from Lemma 4.8 that there exists ǫ ∈ R > such that M ∩ [ c, c + ǫ ] = { c } . Since M is not finitely generated, Proposition 4.3 ensures that f M is dense in R ≥ and, therefore, there exists e c ∈ f M such that 0 < e c < ǫ . Nowtake e c := c + e c . The element e c belongs to f M and satisfies that c < e c < c + ǫ , whence e c ∈ f M \ M . In addition, e c > c > r . As r was taken arbitrarily in R , we obtainthat sup f M \ M = ∞ , as desired. Hence it follows from Proposition 4.5 that c ( M ) isnecessarily empty.For the reverse implication of (2), it suffices to notice that M is finitely generated ifand only if M is isomorphic to a numerical monoid and that, as seen in Example 4.4,the conductor of a numerical monoid is always nonempty. (cid:3) We have seen that if a non-finitely generated monoid is increasing, then it has emptyconductor. We would like to remark that there are non-finitely generated atomicPuiseux monoids that are not increasing and still have empty conductor. The followingexample illustrates this.
Example 4.10.
Consider the Puiseux monoid M generated by the infinite set A := { } ∪ { /p | p ∈ P } . Since 0 is not a limit point of M • , it follows from [17,Proposition 4.5] that M is atomic. Indeed, it is not hard to check that A ( M ) = A . So M is not finitely generated. On the other hand, the fact that 1 is a limit point of M from the right, along with Lemma 4.8, guarantees that M is not an increasing Puiseuxmonoid. In addition, it has been verified in [12, Example 3.9] that c ( M ) is empty. TOMICITY AND DENSITY OF PUISEUX MONOIDS Acknowledgments
The authors would like to thank Felix Gotti for his valuable feedback during thepreparation of this paper.
References [1] A. Assi and P. A. Garc´ıa-S´anchez:
Numerical Semigroups and Applications . New York: Springer-Verlag, 2016.[2] N. R. Baeth and F. Gotti:
Factorizations in upper triangular matrices over information semial-gebras . Available on arXiv: https://arxiv.org/pdf/2002.09828.pdf[3] M. Bras-Amor´os:
Tempered monoids of real numbers, the golden fractal monoid, and the well-tempered harmonic semigroup , Semigroup Forum (2019) 496–516.[4] M. Bras-Amor´os: Increasingly enumerable submonoids of R , Amer. Math. Monthly (to appear).[5] S. T. Chapman, F. Gotti, and M. Gotti: Factorization invariants of Puiseux monoids generatedby geometric sequences , Comm. Algebra (2020) 380–396.[6] S. T. Chapman, F. Gotti, and M. Gotti: When is a Puiseux monoid atomic? , Amer. Math.Monthly (to appear). Available on arXiv: https://arxiv.org/pdf/1908.09227.pdf[7] P. Cohn:
Bezout rings and and their subrings , Proc. Cambridge Philos. Soc. (1968) 251–264.[8] J. Coykendall, D. E. Dobbs, and B. Mullins: On integral domains with no atoms , Comm. Algebra (1999), 5813–5831.[9] J. Coykendall and F. Gotti: On the atomicity of monoid algebras , J. Algebra (2019) 138–151.[10] P. A. Garc´ıa-S´anchez and J. C. Rosales:
Numerical Semigroups , Developments in Mathematics,20, Springer-Verlag, New York, 2009.[11] A. Geroldinger:
Sets of Lengths , Amer. Math. Monthly (2016), 960–988.[12] A. Geroldinger, F. Gotti, and S. Tringali:
On strongly primary monoids, with a focus on Puiseuxmonoids . Available on arXiv: https://arxiv.org/pdf/1910.10270.pdf[13] A. Geroldinger and F. Halter-Koch:
Non-Unique Factorizations: Algebraic, Combinatorial andAnalytic Theory , Pure and Applied Mathematics, vol. 278, Chapman & Hall/CRC, Boca Raton,2006.[14] A. Geroldinger and M. Roitman,
On strongly primary monoids and domains . Available on arXiv:https://arxiv.org/abs/1807.10683.pdf[15] A. Geroldinger and W. Schmid:
A realization theorem for sets of lengths in numerical monoids ,Forum Math. (2018) 1111–1118.[16] F. Gotti: On the atomic structure of Puiseux monoids , J. Algebra Appl. (2017), 1750126.[17] F. Gotti: Increasing positive monoids of ordered fields are FF-monoids , J. Algebra (2019),40–56.[18] F. Gotti:
Puiseux monoids and transfer homomorphisms , J. Algebra (2018), 95–114.[19] F. Gotti and M. Gotti:
Atomicity and boundedness of monotone Puiseux monoids , SemigroupForum (2018), 536–552.[20] A. Grams: Atomic domains and the ascending chain condition for principal ideals . Math. Proc.Cambridge Philos. Soc. (1974), 321–329.[21] P. A. Grillet: Commutative Semigroups , Advances in Mathematics, vol. 2, Kluwer AcademicPublishers, Boston, 2001. MARIA BRAS-AMOROS AND MARLY GOTTI
Departament d’Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Vir-gili, Avinguda dels Pa¨ısos Catalans 26, E-43007 Tarragona, Spain
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