Approximating length-based invariants in atomic Puiseux monoids
aa r X i v : . [ m a t h . A C ] J u l APPROXIMATING LENGTH-BASED INVARIANTS IN ATOMICPUISEUX MONOIDS
HAROLD POLO
Abstract.
A numerical monoid is a cofinite additive submonoid of the nonnegativeintegers, while a Puiseux monoid is an additive submonoid of the nonnegative coneof the rational numbers. Using that a Puiseux monoid is an increasing union ofcopies of numerical monoids, we prove that some of the factorization invariants ofthese two classes of monoids are related through a limiting process. This allows usto extend results from numerical to Puiseux monoids. We illustrate the versatility ofthis technique by recovering various known results about Puiseux monoids. Introduction
A monoid M is atomic provided that every nonunit element can be representedas a product of finitely many irreducibles. If for each nonunit element of M such arepresentation is unique, up to permutation, then M is called a unique factorizationmonoid (UFM). For example, the positive integers with the standard product is a UFMby the Fundamental Theorem of Arithmetic. Factorization theory studies how far is anatomic monoid from being a UFM, and several algebraic invariants has been introducedto quantify this deviation (see [15] and references therein).Numerical monoids, that is, cofinite additive submonoids of the nonnegative integers,have been significantly investigated in the context of factorization theory; much of therecent literature has focused on the computational aspects of their factorization invari-ants (see, for example, [2]). Since numerical monoids are finitely generated, calculatingfactorization invariants in this setting is highly tractable [10]. This motivated the im-plementation of a GAP [13] package, numericalsgps [9], to assist researchers in thearea. Thus, numerical monoids constitute an ideal framework to study factorizationinvariants.Additive submonoids of the nonnegative cone of Q , also called Puiseux monoids,are natural generalizations of numerical monoids. A systematic investigation of thesemonoids started just a few years ago in [16] and, consequently, we do not know muchabout their factorization invariants. The crux of this article is to study the set of lengths(and related factorization invariants) of Puiseux monoids through their representationas increasing unions of copies of numerical monoids. Date : July 21, 2020.2010
Mathematics Subject Classification.
Primary: 20M13; Secondary: 40A05, 20M14.
Key words and phrases. atomic Puiseux monoids, numerical monoids, approximation, factorizationinvariants, sets of lengths, elasticity, set of distances. H. POLO Preliminary
In this section, we introduce the concepts and notation necessary to follow our ex-position. General references for factorization theory can be found in [14].Throughout this article, we let N and N denote the set of positive and nonnegativeintegers, respectively, while we denote by R the set R ∪ {∞} . For nonnegative integers m and n , let J m, n K be the set of integers between m and n , i.e., J m, n K := { k ∈ N | m ≤ k ≤ n } . Given a subset S of the rational numbers, we let S ≥ t denote the set of nonnegativeelements of S that are greater than or equal to t . In the same way we define S >t and S An atomic Puiseux monoid M can be represented as an increasing union of copiesof numerical monoids: the monoid M contains a minimal set of generators, namely A ( M ), by [14, Proposition 1.1.7]. Consequently, given an ordering a , a , . . . of theelements of A ( M ), we have the sequence ( N i ) i ≥ with N i = h a , . . . , a i i for all i ∈ N .Clearly, M = S i ≥ N i and N i is isomorphic to a numerical monoid for each i ∈ N by[19, Theorem 4.2]. This representation has been used to manufacture Puiseux monoidssatisfying certain properties. Consider the following examples. Example 3.1. In [21, Section 6] the authors constructed a bifurcus Puiseux monoid,that is, a Puiseux monoid M satisfying that 2 ∈ L ( x ) for all x ∈ M • \A ( M ). To achieve H. POLO this, take a collection of prime numbers { p j,n | j,n ≥ } such that p j,n ≥ max(13 , j )for all j, n ∈ N and, recursively, define an increasing sequence of finitely generatedPuiseux monoids in the following manner: take N = h / , / i , and assuming that N j − was already defined for some j ∈ N , let x j, , x j, , . . . be the elements of N j − withno length 2 factorization. Then take N j = N j − + (cid:28) x j,n − p j,n , x j,n p j,n (cid:12)(cid:12)(cid:12)(cid:12) n ≥ (cid:29) . Observe that N j provides a length 2 factorization for the elements of N j − that didnot have one before. Now take M = S i ≥ N i . The monoid M is bifurcus; the readercan check the details of the proof in [21, Theorem 6.2]. One of the key features of thisconstruction is that A ( N i ) ⊆ A ( N i +1 ) for every i ∈ N . Example 3.2. In [18] the author proved that there exists a Puiseux monoid without0 as a limit point that has no finite local elasticities. With this purpose, she piecestogether a Puiseux monoid M by creating a strictly increasing sequence of finite subsetsof positive rationals ( A i ) i ≥ satisfying the following three conditions: • d ( A i ) consists of odd prime numbers, • d (max A i ) = max d ( A i ), and • A i minimally generates the Puiseux monoid N i = h A i i .Then the author takes M = S i ≥ N i , where A ( N i ) ⊆ A ( N i +1 ) ⊆ A ( M ) and prove that( ρ ( N i )) i ≥ is an increasing sequence that does not stabilize. Since A ( N i ) ⊆ A ( M ) foreach i ∈ N , it follows that ρ ( M ) = ∞ . For details see [18, Proposition 3.6].This representation of Puiseux monoids can help us not only to provide sophisticatedexamples but also to study some factorization invariants in these monoids. Definition 3.3. Let ( M i ) i ≥ be an increasing sequence of atomic Puiseux monoids. Wesay that ( M i ) i ≥ is an approximation of the Puiseux monoid M = S i ≥ M i providedthat A ( M i ) ⊆ A ( M i +1 ) for each i ∈ N . If M i is finitely generated for every i ∈ N thenwe call ( M i ) i ≥ a numerical approximation of M . Remark 3.4. Given an approximation ( M i ) i ≥ of a Puiseux monoid M , it is not hardto see that M is atomic with A ( M ) = S i ≥ A ( M i ).We prove that, given an approximation of a Puiseux monoid, we can compute itssets of lengths and related factorization invariants by “passing to the limit” in a sensethat will become clear soon. Using this approach we can provide alternative proofs tosome known results about the sets of lengths of Puiseux monoids. Theorem 3.5. Let M be a Puiseux monoid with an approximation ( M i ) i ≥ , and let x be an element of M . Then, for some j ∈ N , the following statements hold: (1) Z M ( x ) = S i ≥ j Z M i ( x ) and Z ( M ) = S i ≥ Z ( M i ) . PPROXIMATING LENGTH-BASED INVARIANTS L M ( x ) = S i ≥ j L M i ( x ) . (3) ρ M ( x ) = lim i ρ M i + j ( x ) and ρ ( M ) = lim i ρ ( M i ) . (4) ρ m ( M ) = lim i ρ m ( M i ) for each m ∈ N .Proof. Let j, r, s ∈ N such that x ∈ M j and j ≤ r ≤ s . Since A ( M r ) ⊆ A ( M s ), theinclusion Z M r ( x ) ⊆ Z M s ( x ) holds. Now if z ∈ Z M i ( x ) for some i ∈ N then z ∈ Z M ( x )by Remark 3.4. Conversely, if z = a + · · · + a n ∈ Z M ( x ) with a , . . . , a n ∈ A ( M )then there exists k ∈ N ≥ j such that a i ∈ A ( M k ) for each i ∈ J , n K . Consequently, z ∈ Z M k ( x ). Hence Z M ( x ) = S i ≥ j Z M i ( x ). For all y ∈ M , let j ( y ) ∈ N such that y ∈ M j ( y ) . Thus, Z ( M ) = [ y ∈ M Z M ( y ) = [ y ∈ M [ i ≥ j ( y ) Z M i ( y ) = [ i ≥ Z ( M i ) , from which (1) follows. It is easy to see that (2) readily follows from (1).If x = 0 then the first part of (3) clearly follows, so there is no loss in assuming that x = 0. Since L M r ( x ) ⊆ L M s ( x ) ⊆ L M ( x ), the inequalities ρ M r ( x ) ≤ ρ M s ( x ) ≤ ρ M ( x )hold, which implies that lim i ρ M i + j ( x ) exists (in R ) and lim i ρ M i + j ( x ) ≤ ρ M ( x ). Forthe reverse inequality, note that if L M ( x ) is unbounded then ρ M ( x ) = ∞ . In thiscase, for each n ∈ N , there exists z = a + · · · + a l ∈ Z M ( x ) with a , . . . , a l ∈ A ( M )satisfying that l > n . By virtue of (2), there exists k ∈ N ≥ j such that l ∈ L M k ( x ).Since L M i + j ( x ) ⊆ L M i + j +1 ( x ) for each i ∈ N , we have lim i ρ M i + j ( x ) = ∞ . On the otherhand, if L M ( x ) is bounded then, for some h ∈ N ≥ j , we have ρ M ( x ) = sup L M ( x )inf L M ( x ) = sup S i ≥ j L M i ( x )inf S i ≥ j L M i ( x ) = max L M h ( x )min L M h ( x ) = ρ M h ( x ) ≤ lim i →∞ ρ M i + j ( x ) . Next we prove that ρ ( M ) = lim i ρ ( M i ). We already established that, for each i ∈ N ,the inequality ρ M i ( y ) ≤ ρ M i +1 ( y ) holds for all y ∈ M i . Consequently, ρ ( M i ) ≤ ρ ( M i +1 )for each i ∈ N which, in turn, implies that lim i ρ ( M i ) exists (in R ). By definition, ρ ( M ) ≥ ρ M ( y ) for all y ∈ M . Now fix j ∈ N , and let y ′ ∈ M j . Since ρ M ( y ′ ) ≥ ρ M j ( y ′ ),the inequality ρ ( M ) ≥ ρ M j ( y ′ ) holds for all y ′ ∈ M j , which implies that ρ ( M ) ≥ ρ ( M j ).This, in turn, implies that ρ ( M ) ≥ lim i ρ ( M i ). To prove the reverse inequality, observethat, for all y ∈ M , we have ρ M ( y ) = lim i ρ M i + j ( y ) ( y ) ≤ lim i ρ ( M i ). This implies that ρ ( M ) ≤ lim i ρ ( M i ), and (3) holds.For all i ∈ N , the inclusions U m ( M i ) ⊆ U m ( M i +1 ) ⊆ U m ( M ) hold. Consequently,sup U m ( M i ) ≤ sup U m ( M i +1 ) ≤ sup U m ( M ) which, in turn, implies that lim i ρ m ( M i )exists (in R ) and lim i ρ m ( M i ) ≤ ρ m ( M ). Now if U m ( M ) is unbounded then for each N ∈ N there exist x ∈ M and z, z ′ ∈ Z M ( x ) such that | z | > N and | z ′ | = m . Sincethere exists j ∈ N such that z, z ′ ∈ Z M j ( x ), the inequality ρ m ( M j ) > N holds. Thisimplies that lim i ρ m ( M i ) = ∞ . Then there is no loss in assuming that k := sup U m ( M )is a positive integer. Let x ∈ M such that | z | = k and | z ′ | = m for some z, z ′ ∈ Z M ( x ).Since z, z ′ ∈ Z M j ( x ) for some j ∈ N , our argument follows. (cid:3) H. POLO Corollary 3.6. [21, Theorem 3.2] Let M be an atomic Puiseux monoid. If is a limitpoint of M • then ρ ( M ) = ∞ . Otherwise, ρ ( M ) = sup A ( M )inf A ( M ) .Proof. Let ( N i ) i ≥ be a numerical approximation of M . If 0 is a limit point of M • then,for each n ∈ N , there exists j ∈ N such that ρ ( N j ) > n by [7, Theorem 2.1], whichimplies that lim i ρ ( N i ) = ∞ since ( ρ ( N i )) i ≥ is nondecreasing. Now if 0 is not a limitpoint of M • then ρ ( M ) = lim i →∞ ρ ( N i ) = lim i →∞ max A ( N i )min A ( N i ) = sup A ( M )inf A ( M ) , where the second equality follows from [7, Theorem 2.1]. (cid:3) Corollary 3.7. [21, Theorem 3.4] Let M be an atomic Puiseux monoid satisfying that ρ ( M ) < ∞ . Then the elasticity of M is accepted if and only if A ( M ) has both amaximum and a minimum.Proof. Let ( N i ) i ≥ be a numerical approximation of M . To tackle the direct implication,note that for some x ∈ M , j ∈ N , and L, l ∈ L M ( x ) we havesup A ( M )inf A ( M ) = ρ ( M ) = ρ M ( x ) = Ll = ρ N j ( x ) = max A ( N j )min A ( N j ) , where the last equality follow from [7, Theorem 2.1]. The reverse implication followsfrom [14, Theorem 3.1.4] and the fact that, for some j ∈ N , the monoid N j containsthe minimum and maximum of A ( M ). (cid:3) Corollary 3.8. [18, Proposition 3.1] Let M be an atomic Puiseux monoid. If M contains a stable atom a ∈ A ( M ) then ρ k ( M ) is infinite for all sufficiently large k .Proof. Let ( N i ) i ≥ be a numerical approximation of M , and suppose without lossof generality that a ∈ N . For each j ∈ N , there exists k ∈ N such that theinequality ρ d ( a ) ( N j + k ) > ρ d ( a ) ( N j ) holds since N j is finitely generated. Therefore,lim i ρ d ( a ) ( N i ) = ∞ . By Theorem 3.5, we have ρ d ( a ) ( M ) = ∞ . Our argument followsafter [14, Proposition 1.4.2]. (cid:3) Set of Distances It is straightforward to construct a Puiseux monoid M with an approximation( M i ) i ≥ such that ∆( M ) = S i ≥ ∆( M i ). Consequently, the approach we used in Theo-rem 3.5 to compute invariants like the set of lengths is not going to work for the set ofdistances. However, using limits of sets we can obtain a result similar to Theorem 3.5. Definition 4.1. Let ( S i ) i ≥ be a sequence of sets, and let lim inf i S i and lim sup i S i bethe sets lim inf i →∞ S i := [ i ≥ \ j ≥ i S j and lim sup i →∞ S i := \ i ≥ [ j ≥ i S i . We say that lim i S i exists and is equal to lim inf i S i provided that lim inf i S i = lim sup i S i . PPROXIMATING LENGTH-BASED INVARIANTS S i ) i ≥ is an increasing sequence of sets then lim i S i = S i ≥ S i as the reader can easilyprove. Proposition 4.2. Let M be a Puiseux monoid with an approximation ( M i ) i ≥ , and let x be an element of M . Then ∆ M ( x ) ⊆ lim inf i ∆ M i ( x ) and ∆( M ) ⊆ lim inf i ∆( M i ) .Proof. Let d ∈ ∆ M ( x ). Then there exist factorizations z, z ′ ∈ Z M ( x ) satisfying that | z ′ | − | z | = d and [ | z | , | z ′ | ] ∩ L M ( x ) = {| z | , | z ′ |} . Let k ∈ N such that z, z ′ ∈ Z M k ( x ). Byvirtue of Theorem 3.5, we have that d ∈ ∆ M h ( x ) for all h ∈ N ≥ k which, in turn, impliesthat d ∈ T j ≥ k ∆ M j ( x ). Then d ∈ lim inf i ∆ M i ( x ). Finally, let d ∈ ∆( M ). By definition,there exists x ∈ M • such that d ∈ ∆ M ( x ). As we already showed, d ∈ T j ≥ k ∆ M j ( x )for some k ∈ N . Consequently, d ∈ T j ≥ k ∆( M j ), from which our result follows. (cid:3) Proposition 4.2 can be useful when analyzing the set of lengths of particular classesof atomic Puiseux monoids. Consider the following examples. Example 4.3. Let r ∈ Q < such that the rational cyclic monoid over r , that is, S r := h r n | n ∈ N i , is atomic. Then n ( r ) > i ∈ N , andconsider the numerical monoid N i = h n ( r ) i , n ( r ) i − d ( r ) , . . . , d ( r ) i i . By virtue of [22,Corollary 20], we have ∆( N i ) = { d ( r ) − n ( r ) } . It is not hard to see that ( d ( r ) − i N i ) i ≥ isa numerical approximation of S r . Therefore, ∆( S r ) ⊆ { d ( r ) − n ( r ) } by Proposition 4.2.Following a similar reasoning we obtain that if r > S r is atomic then the inclusion∆( S r ) ⊆ { n ( r ) − d ( r ) } holds. This result was first proved in [6, Theorem 3.3]. Example 4.4. Let B be a nonempty subset of Q > \ N such that for all b, b ′ ∈ B with b = b ′ we have n ( b ) > 1, gcd( d ( b ) , d ( b ′ )) = 1, and | n ( b ) − d ( b ) | = | n ( b ′ ) − d ( b ′ ) | . Set M B := h b n | b ∈ B , n ∈ N i . Now given an ordering b , b , . . . of the elements of B , let B i = { b , . . . , b i } and set M B i := h b n | b ∈ B i , n ∈ N i for each i ∈ N . The sequence( M B i ) i ≥ is an approximation of M B by [23, Proposition 3.5]. Moreover, for each i ∈ N ,∆( M B i ) = {| n ( b ) − d ( b ) |} by [23, Theorem 4.9]. Therefore, ∆( M B ) ⊆ {| n ( b ) − d ( b ) |} by Proposition 4.2. Remark 4.5. Example 4.4 extends part (2) of [23, Theorem 4.9] to a larger class ofPuiseux monoids.The next example shows that, in general, ∆( M ) = lim inf i ∆( M i ). Example 4.6. Consider the rational cyclic monoid S r with r ∈ Q > \ N . For each i ∈ N , set M i := (cid:10)(cid:8) r k | k ∈ N (cid:9) ∪ (cid:8) r j − | j ∈ J , i K (cid:9)(cid:11) . It is not hard to prove that ( M i ) i ≥ is an approximation of S r . Now fix i ∈ N , and let x i = n ( r ) r i ∈ M i . Clearly, z = n ( r ) r i and z ′ = d ( r ) r i +2 are two factorizations of x i in M i . Claim 1. z ′ = d ( r ) r i +2 ∈ Z M i ( x i ) is the factorization of minimum length of x i in M i . H. POLO Proof. Let z ′′ = P nk =0 c k r s k ∈ Z M i ( x i ) with coefficients c , . . . , c n ∈ N and exponents s , . . . , s n ∈ { k | k ∈ N } ∪ { j − | j ∈ J , i K } , and assume by contradiction that z ′′ isa factorization of minimum length of x i in M i satisfying that z ′′ = z ′ . There is no lossin assuming that s l < s r for l < r , [ r s l , r s l +1 ] ∩ A ( M i ) = { r s l , r s l +1 } for all l ∈ J , n − K and s t = 2 i + 2 for some t ∈ J , n K . Note that c k < n ( r ) s k +1 − s k for each k ∈ J , n K ;otherwise, using the transformation n ( r ) s k +1 − s k r s k = d ( r ) s k +1 − s k r s k +1 we can generatea new factorization z ∗ ∈ Z M i ( x i ) such that | z ∗ | < | z ′′ | , which is a contradiction. Nowlet m be the smallest nonnegative integer such that c m = 0, and consider the equation(4.1) n X k = m c k r s k = d ( r ) r i +2 . If m < t then after clearing denominators in Equation (4.1) we generate a contradictionwith the fact that c m < n ( r ) s m +1 − s m . We obtain a similar contradiction for the casewhere m ≥ t as the reader can verify. Therefore, there exists exactly one factorizationof minimum length of x i in M i , namely z ′ . (cid:3) Now let z ∗ = P nk =0 c k r s k ∈ Z M i ( x i ) with coefficients c , . . . , c n ∈ N and exponents s , . . . , s n ∈ { k | k ∈ N } ∪ { j − | j ∈ J , i K } . Suppose, without loss of generality,that s l < s r for l < r and [ r s l , r s l +1 ] ∩ A ( M i ) = { r s l , r s l +1 } for every l ∈ J , n − K . Notethat in the proof of Claim 1, we established that if c k < n ( r ) s k +1 − s k for each k ∈ J , n K then z ∗ = z ′ . Claim 2. If | z ∗ | < | z | = n ( r ) then z ∗ = z ′ . Proof. If c k < n ( r ) s k +1 − s k for each k ∈ J , n K then we are done by our previous obser-vation. By contradiction, assume that z ∗ = z ′ . Using the transformation(4.2) n ( r ) s k +1 − s k r s k = d ( r ) s k +1 − s k r s k +1 we can generate from z ∗ a new factorization z ∈ Z M i ( x i ) such that | z | < | z ∗ | . Theneither z = z ′ or we can again apply the transformation (4.2) to obtain a new factoriza-tion z ∈ Z M i ( x i ) such that | z | < | z | , and so on. This procedure stops since there isno strictly decreasing sequence of nonnegative integers. Then there exist factorizations z ∗ = z , z , . . . , z m = z ′ such that | z j | > | z j +1 | for every j ∈ J , m − K . It shouldbe noted that the transformation (4.2) increases the exponent of r , which means that c k = 0 for all s k > i + 2, where k ∈ J , n K . This implies that at some point in theaforementioned procedure we applied the transformation n ( r ) r i = d ( r ) r i +2 , but thiscontradicts that | z ∗ | < | z | = n ( r ) . (cid:3) Because of Claim 2, n ( r ) − d ( r ) ∈ ∆( M i ) for every i ∈ N . Consequently, we havethat n ( r ) − d ( r ) ∈ lim inf i ∆( M i ). However, we know that ∆( S r ) = { n ( r ) − d ( r ) } by[6, Corollary 3.4]. Therefore, ∆( S r ) = lim inf i ∆( M i ). 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