# Maria Bras-Amorós's research while affiliated with Universitat Rovira i Virgili and other places

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## Publications (80)

We address the problem of constructing large circulant networks with given degree and diameter, and efficient routing schemes. First we discuss the theoretical upper bounds and their asymptotics. Then we apply concepts and tools from the change-making problem to efficient routing in circulant graphs. With these tools we investigate some of the fami...

A flag of codes C0 ⊊ C1 ⊊ ... ⊊ Cs ⊆ Fnq is said to satisfy the isometry-dual property if there exists x ∈ (F*q)n such that the code Ci is x-isometric to the dual code C⊥s-i for all i = 0, ..., s. For P and Q rational places in a function field F, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic ge...

We give a graphical reinterpretation of the seeds algorithm to explore the tree of numerical semigroups. We then exploit the seeds algorithm to find all the Eliahou semigroups of genus up to 65. Since all these semigroups satisfy the Wilf conjecture, this shows that the Wilf conjecture holds up to genus 65.

For Reed-Solomon codes, the key equation relates the syndrome polynomial---computed from the parity check matrix and the received vector---to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives t...

We determine the Weierstrass semigroup $H(P_\infty,P_1,\ldots,P_m)$ at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced by Tafazolian, Teher\'an-Herrera, and Torres. Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with better relative paramete...

In this study, we present the notion of the quasi-ordinarization transform of a numerical semigroup. The set of all semigroups of a fixed genus can be organized in a forest whose roots are all the quasi-ordinary semigroups of the same genus. This way, we approach the conjecture on the increasingness of the cardinalities of the sets of numerical sem...

Let ${\mathbb F}_q$ be the finite field with $q$ elements and let ${\mathbb N}$ be the set of non-negative integers. A flag of linear codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s$ is said to have the {\it isometry-dual property} if there exists a vector ${\bf x}\in (\mathbb{F}_q^*)^n$ such that $C_i={\bf x} \cdot C_{s-i}^\perp$, where...

We present the quasi-ordinarization transform of a numerical semigroup. This transform will allow to organize all the semigroups of a given genus in a forest rooted at all quasi-ordinary semigroups of that genus. This construction provides an alternative approach to the conjecture on the increasingness of the number of numerical semigroups of each...

A Puiseux monoid is an additive submonoid consisting of non-negative rationals. Although the operation of addition is continuous with respect to the standard topology, the set of irreducibles of a Puiseux monoid is, in general, difficult to describe. Here, we use topological density to understand how much a Puiseux monoid, as well as its set of irr...

Anonymization for privacy-preserving data publishing, also known as statistical disclosure control (SDC), can be viewed under the lens of the permutation model. According to this model, any SDC method for individual data records is functionally equivalent to a permutation step plus a noise addition step, where the noise added is marginal, in the se...

Consider a complete flag \(\{0\} = C_0< C_1< \cdots < C_n = \mathbb {F}^n\) of one-point AG codes of length n over the finite field \(\mathbb {F}\). The codes are defined by evaluating functions with poles at a given point Q in points \(P_1,\dots ,P_n\) distinct from Q. A flag has the isometry-dual property if the given flag and the corresponding d...

A flag of codes $C_0 \subsetneq C_1 \subsetneq \cdots \subsetneq C_s \subseteq {\mathbb F}_q^n$ is said to satisfy the {\it isometry-dual property} if there exists ${\bf x}\in (\mathbb{F}_q^*)^n$ such that the code $C_i$ is {\bf x}-isometric to the dual code $C_{s-i}^\perp$ for all $i=0,\ldots, s$. For $P$ and $Q$ rational places in a function fiel...

We determine the Weierstrass semigroup \(H(P_\infty ,P_1,\ldots ,P_m)\) at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced in Tafazolian et al. (J Pure Appl Algebra 220(3):1122–1132, 2016). Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain AG codes with...

A Puiseux monoid is a submonoid of $(\mathbb{Q},+)$ consisting of nonnegative rational numbers. Although the operation of addition is continuous with respect to the 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 how much a Puiseux monoid,...

Anonymization for privacy-preserving data publishing, also known as statistical disclosure control (SDC), can be viewed under the lens of the permutation model. According to this model, any SDC method for individual data records is functionally equivalent to a permutation step plus a noise addition step, where the noise added is marginal, in the se...

We analyze the set of increasingly enumerable additive submonoids of R, for instance, the set of logarithms of the positive integers with respect to a given base. We call them ω-monoids. The ω-monoids for which consecutive elements become arbitrarily close are called tempered monoids. This is, in particular, the case for the set of logarithms. We s...

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabil...

For a numerical semigroup, we encode the set of primitive elements that are larger than its Frobenius number and show how to produce in a fast way the corresponding sets for its children in the semigroup tree. This allows us to present an efficient algorithm for exploring the tree up to a given genus. The algorithm exploits the second nonzero eleme...

This paper presents a new way to view the key equation for decoding Reed–Solomon codes that unites the two algorithms used in solving it—the Berlekamp–Massey algorithm and the Euclidean algorithm. A new key equation for Reed–Solomon codes is derived for simultaneous errors and erasures decoding using the symmetry between polynomials and their recip...

This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way. The sequence of physical harmonics is an increasingly enumerable submonoid of \(({{\mathbb {R}}}^+,+)\) whose pairs of consecutive terms get arbitrarily close as they grow. T...

Consider a complete flag $\{0\} = C_0 < C_1 < \cdots < C_n = \mathbb{F}^n$ of one-point AG codes of length $n$ over the finite field $\mathbb{F}$. The codes are defined by evaluating functions with poles at a given point $Q$ in points $P_1,\dots,P_n$ distinct from $Q$. A flag has the isometry-dual property if the given flag and the corresponding du...

We analyze the set of increasingly enumerable additive submonoids of R, for instance, the set of logarithms of the positive integers with respect to a given base. We call them !-monoids. The !-monoids for which consecutive elements become arbitrarily close are called tempered monoids. This is, in particular, the case for the set of logarithms. We s...

A numerical semigroup is a subset of N containing 0, closed under addition and with finite complement in N. An important example of numerical semigroup is given by the Weierstrass semigroup at one point of a curve. In the theory of algebraic geometry codes, Weierstrass semigroups are crucial for defining bounds on the minimum distance as well as fo...

Because of their importance in applications and their quite simple definition, Reed-Solomon codes can be explained in any introductory course on coding theory. However, decoding algorithms for Reed-Solomon codes are far from being simple and it is difficult to fit them in introductory courses for undergraduates. We introduce a new decoding approach...

R-moulds of numerical semigroups are defined as increasing sequences of real numbers whose discretizations may give numerical semigroups. The ideal sequence of musical harmonics is an R-mould and discretizing it is equivalent to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplic...

For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of seed by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper i...

A tuple a = (a 1, . . . , an) of positive real numbers is said to be equiangular if there is an equiangular polygon with consecutive side lengths a 1, . . . , an. It is well known that a is equiangular if and only if the polynomial a(x) = a 1 + a 2x + · · · + an– 1x n–2 + anxn–1 vanishes at e 2π/n i . Here we dispense with complex numbers and borro...

The peer-to-peer user-private information retrieval (P2P UPIR) protocol is an anonymous database search protocol in which the users collaborate in order to protect their privacy. This collaboration can be modelled by a combinatorial configuration. This chapter surveys currently available results on how to choose combinatorial configurations for P2P...

A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then, the result is used, through the so-called Feng–Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes. This is further developed for Hermitian codes and the c...

The Geil–Matsumoto bound conditions the number of rational places of a function field in terms of the Weierstrass semigroup of any of the places. Lewittes’ bound preceded the Geil–Matsumoto bound and it only considers the smallest generator of the numerical semigroup. It can be derived from the Geil–Matsumoto bound and so it is weaker. However, for...

We extend the list of known linear patterns admitted by numerical semigroups associated with combinatorial configurations. This is done through the construction of configurations from combinations of several smaller configurations. These results may be used to construct configurations with certain parameters, and therefore contribute with answers t...

We prove that the numerical semigroups associated to the combinatorial configurations satisfy a family of linear, non-homogeneous, symmetric patterns. We use these patterns to prove an upper bound of the conductor and we also give an upper bound of the multiplicity. Also, we compare bounds of the conductor of numerical semigroups associated to bala...

Patterns on numerical semigroups are multivariate linear polynomials, and
they are said to be admissible if there exists a numerical semigroup such that
evaluated at any nonincreasing sequence of elements of the semigroup gives
integers belonging to the semigroup. In a first approach, only homogeneous
patterns where analized. In this contribution w...

A unique decoding algorithm for general AG codes, namely multipoint evaluation codes on algebraic curves, is presented. It is a natural generalization of the previous decoding algorithm which was only for one-point AG codes. As such, it retains the same advantages of fast speed, regular structure, and direct message recovery. Upon this generalizati...

A numerical semigroup is said to be ordinary if it has all its gaps in a row.
Indeed, it contains zero and all integers from a given positive one. One can
define a simple operation on a non-ordinary semigroup, which we call here the
ordinarization transform, by removing its smallest non-zero non-gap (the
multiplicity) and adding its largest gap (th...

A numerical semigroup is said to be ordinary if it has all its gaps in a row. Indeed, it contains zero and all integers from a given positive one. One can define a simple operation on a non-ordinary semigroup, which we call here the ordinarization transform, by removing its smallest non-zero non-gap (the multiplicity) and adding its largest gap (th...

The way an author or a group of authors are cited tells more about the real impact of their work than authorship and collaborations. Indeed, the connections within the scientific community can be more accurately elicited from the co-citation graph than from the collaboration graph. We suggest some indices that can be drawn from the co-citation grap...

We present a unique decoding algorithm of algebraic geometry codes on plane
curves, Hermitian codes in particular, from an interpolation point of view. The
algorithm successfully corrects errors of weight up to half of the order bound
on the minimum distance of the AG code. The decoding algorithm is the first to
combine some features of the interpo...

It is proved that a numerical semigroup can be associated to the triangle-free (r,k)-configurations, and some results on existence are deduced. For example it is proved that for any r,k≥2 there exists infinitely many (r,k)-configurations. Most proofs are given from a graph theoretical point of view, in the sense that the configurations are represen...

The popular h-index used to measure scientific output can be described in terms of a pool of evaluated objects (the papers), a quality function on the evaluated objects (the number of citations received by each paper) and a sentencing line crossing the origin, whose intersection with the graph of the quality function yields the index value (in the...

User-private information retrieval (UPIR) is the art of retrieving information without telling the information holder who you are. UPIR is sometimes called anonymous keyword search. This article discusses a UPIR protocol in which the users form a peer-to-peer network over which they collaborate in protecting the privacy of each other. The protocol...

The h-index by Hirsch[1] has recently earned a lot of popularity in bibliometrics, being echoed in Nature and implemented
in the Web of Science bibliometric database. Previous indicators were the total number of papers or the total number of citations.
Following the widely accepted idea that not all papers should count equally, the h-index counts o...

User-private information retrieval systems should protect the user’s anonymity when performing queries against a database, or they should limit the servers capacity of profiling users. Peer-to-peer user-private information retrieval (P2P UPIR) supplies a practical solution: the users in a group help each other in doing their queries, thereby preser...

Private information retrieval (PIR) is normally modeled as a game between two players: a user and a database. The user wants to retrieve some item from the database without the latter learning which item is retrieved. Most current PIR protocols are ill-suited to provide PIR from a search engine or large database: (i) their computational complexity...

We introduce a new sequence τ associated to a numerical semigroup similar to the ν sequence used to define the order bound on the minimum distance and
to describe the Feng–Rao improved codes. The new sequence allows a nice description of the optimal one-point codes correcting
generic errors and to compare them with standard codes and with the Feng–...

The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. In this article an alternative version of the key equation and a new way to use the Euclidean algorithm to solve it are presented, which yield the Berleka...

A (v,b,r,k) combinatorial configuration is a (r,k)-biregular bipartite graph with v vertices on the left and b vertices on the right and with no cycle of length 4. Combinatorial configurations have become very important for some cryptographic applications to sensor networks and to peer-to-peer communities. Configurable tuples are those tuples (v,b,...

The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. This article presents a new version of the key equation and a way to use the Euclidean algorithm to solve it. A straightforward reorganization of the algo...

The most successful method to obtain lower bounds for the minimum distance of an algebraic geometric code is the order bound, which generalizes the Feng-Rao bound. We provide a significant extension of the bound that improves the order bounds by Beelen and by Duursma and Park. We include an exhaustive numerical comparison of the different bounds fo...

This book constitutes the refereed proceedings of the 18th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-18, held in Tarragona, Spain, in June 2009.
The 22 revised full papers presented together with 7 extended absstracts were carefully reviewed and selected from 50 submissions. Among the subject...

A (v,b,r,k) combinatorial configuration is a (r,k)-biregular bipartite graph
with v vertices on the left and b vertices on the right and with no cycle of
length 4. Combinatorial configurations have become very important for some
cryptographic applications to sensor networks and to peer-to-peer communities.
Configurable tuples are those tuples (v,b...

We consider the unbounded integer grid and the digitized version of the straight line y = �x + �, with �,� 2 R being the set of points (i, (�i + �)),i 2 Z, where (·) is the integer rounding operator ((x) 0.5 � x < (x) + 0.5). We address the problem of counting the number of points in the integer grid in which two digitized straight lines overlap ea...

In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which of the two types takes place in particular for well-known classes of semigroups. Also we study the question o...

Steganographic file systems are file systems where the location and even the existence of files are unknown to the users not
having stored them. If the file system can be written to by several users, a user may inadvertently damage the files stored
by other users. In this paper, solutions to the collision problem are proposed which rely on error-co...

For Reed-Solomon codes, the key equation relates the syndrome polynomial—computed from the parity check matrix and the received vector—to two unknown polynomials, the locator and the evaluator. The roots of the locator polynomial identify the error positions. The evaluator polynomial, along with the derivative of the locator polynomial, gives the e...

Private information retrieval (PIR) is normally modeled as a game between two players: a user and a database. The user wants
to retrieve some item from the database without the latter learning which item. Most current PIR protocols are ill-suited
to provide PIR from a search engine or large database: i) their computational complexity is linear in t...

This article is focused on some variations of Reed–Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp–Massey–Sakata algorithm with majority voting. Explicit formulas for the redundancies of the new codes are given.

In this work we study some objects describing the addition behavior of a numerical semigroup and we prove that they uniquely determine the numerical semigroup. We then study the case of Arf numerical semigroups and find some specific results. Résumé (Comportement de l'addition dans un semi-groupe numérique). — Dans ce travail, nousetudions des obje...

Lower and upper bounds are given for the number n(g) of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n(g) is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound...

We conjecture a Fibonacci-like property on the number of numerical semigroups of a given genus. Moreover we conjecture that
the associated quotient sequence approaches the golden ratio. The conjecture is motivated by the results on the number of
semigroups of genus at most50. The Wilf conjecture has also been checked for all numerical semigroups wi...

Improvements to code dimension of evaluation codes, while maintaining a fixed decoding radius, were discovered by G.-L. Feng and T. R. N. Rao [IEEE Trans. Inf. Theory 41, No. 6, Pt. 1, 1678–1693 (1995; Zbl 0866.94025)], and nicely described in terms of an order function by T. Høholdt et al. [in: Handbook of coding theory. Amsterdam: Elsevier. 871–9...

We consider a generalization of the codes defined by norm and trace functions on finite fields introduced by Olav Geil. The codes in the new family still satisfy Geil's duality properties stated for normtrace codes. That is, it is easy to find a minimal set of parity checks guaranteeing correction of a given number of errors, as well as the set of...

Garcia and Stichtenoth discovered a tower of function fields that meets the Drinfeld-Vladut bound on the ratio of the number of points to the genus. For this tower, Pellikaan, Stichtenoth, and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this correspondence, a nonr...

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geom...

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single
point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes
and Reed–Muller codes. Given an $${\mathbb{F}}_q$$-algebra A, an order function $$\rho$$ on A and give...

This correspondence is a short extension to the previous article Bras-Amoroacutes, 2004. In that work, some results were given on one-point codes related to numerical semigroups. One of the crucial concepts in the discussion was the so-called nu-sequence of a semigroup. This sequence has been used in the literature to derive bounds on the minimum d...

We introduce square diagrams that represent numerical semigroups and we obtain an injection from the set of numerical semigroups into the set of Dyck paths.

We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary subsets of monomials, and in identifying when the dual codes are also obtained by evaluating monomials. We the...

Garcia and Stichtenoth discovered two towers of function fields that meet the Drinfeld-Vl\u{a}du\c{t} bound on the ratio of the number of points to the genus. For one of these towers, Garcia, Pellikaan and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this article,...

This article is focused on some variations of Reed-Muller codes that
yield improvements to the rate for a prescribed decoding performance
under the Berlekamp-Massey-Sakata algorithm with majority voting.
Explicit formulas for the redundancies of the new codes are given.

Summary form only given. We adapted shape-adaptive coding and the BISK algorithm to new sign and refinement encoders, with the novelty of encoding separately the refinement bits for each set of coefficients having the same prefix. This allows the algorithm to capitalize the refinement redundancy among each of these sets. The proposed sign and refin...

We introduce the notion of pattern for numerical semigroups, which allows us to generalize the definition of Arf numerical semigroups. In this way infinitely many other classes of numerical semigroups are defined giving a classification of the whole set of numerical semigroups. In particular, all semigroups can be arranged in an infinite non-stabil...

Sakata’s generalization of the Berlekamp–Massey algorithm applies to a broad class of codes defined by an evaluation map on an order domain. In order to decode up to the minimum distance bound, Sakata’s algorithm must be combined with the majority voting algorithm of Feng, Rao and Duursma. This combined algorithm can often decode far more than (d
m...

A shape-adaptive search is defined based on the BISK scheme and it is applied to sign encoding and magnitude refinement of
images. It can be generalized to a complete bitplane encoder whose performance is comparable to that of other state-of-the-art
encoders.

The Set Partitioning in Hierarchical Trees (SPIHT) is a well known lossy to lossless high performance embedded bitplane image coding algorithm which uses scalar quantization and zero-trees of transformed bidimensional (2-D) images and bases its performance on the redundancy of the significance of the coefficients in these subband hierarchical trees...

We introduce a new class of numerical semigroups, which we call the class of acute semigroups and we prove that they generalize symmetric and pseudosymmetric numerical semigroups, Arf numerical semigroups, and the semigroups generated by an interval. For a numerical semigroup Λ={λ<sub>0</sub><λ<sub>1</sub><...}, denote ν<sub>i</sub>=#{j|λ<sub>i</su...

Following the ideas of G.-L Feng and T. R. N. Rao [IEEE Trans. Inf. Theory 41, 1678-1693 (1995; Zbl 0866.94025)] and of M. E. O’Sullivan [Decoding of Hermitian codes: Beyond the minimum distance bound. Preprint (2001); A generalization of the Berlekamp-Massey-Sakata algorithm. Preprint (2001)], we present new improvements on evaluation codes. Some...

## Citations

... A simpler implementation, the clockwise routing algorithm developed for use in ring-type circulants, does not guarantee optimal routes. In addition to the routing algorithms mentioned in [17], there are many different implementations of routing algorithms developed at different times by a number of authors for different classes of circulant networks [18][19][20][21][22][23][24]. The existing algorithms for routing in circulant topologies have both advantages and disadvantages and can be used depending on the task. ...

... , g e ; cf. [2]), e ≥ g 1 3 (cf. [6]), when g 1 is large enough and its prime factors are not smaller than g 1 e (cf. ...

... Positive monoids that are increasingly generated have been studied in [7,9,24,27]. On the other hand, positive semirings (i.e., positive monoids closed under multiplication) have been studied in [4,5], while the special case of positive monoids generated by a geometric sequence (necessarily positive semirings) have been studied in [14,16]. ...

Reference: Atomicity of Positive Monoids

... , P m ), and consequently the Weierstrass semigroup H (P ∞ , P 1 , . . . , P m ), at m + 1 points on the curves X a,b,n,s and Y n,s for arbitrary s ≥ 1, generalizing the results in [5]. ...

... Natural increasing sequences of one-point codes are obtained by varying a ≥ 0 in the divisor G = aP . These sequences of codes are analyzed in [4,7]. For two-points codes one can obtain an increasing sequence of codes by fixing β ∈ N and varying a ≥ 0 in the divisor G = aP +βQ, as in the reference [3], or one can extend it to any a ∈ Z. ...

... This method has a minor loss of information and delays time while preserving data privacy. J. Domingo-Ferrer et al. [16] redefined trust and data utility, tested them on a permutation model, and evaluated existing anonymization methods against new metrics, weighing information loss against the risk of privacy leakage. Z. G. Zhou et al. [17] proposed a re-anonymity architecture that released the generated Bayesian network rather than the data itself and optimized the excessive distortion of a specific feature attribute. ...

... For example, Grams [17] used Puiseux monoids (i.e., additive submonoids of Q ≥0 ) to refute Cohn's assertion ([7, Proposition 1.1]) that every atomic integral domain satisfies the ascending chain condition on principal ideals. More recently, Bras-Amorós [4] highlighted connections between positive monoids and music theory, while Coykendall and Gotti [9] employed Puiseux monoids to tackle a question posed by Gilmer almost four decades ago in [14, page 189]. The aim of the present article is to study the positive monoids that satisfy the finite factorization property. ...

... The semigroup tree has been used to test Wilf's conjecture up to a given genus, for instance up to genus 60 by Fromentin and Hivert in [20]. Some improvements have been developed to explore the semigroup tree both from a computational point of view (for instance in [8,20]) and from a theoretical point of view, depending on the properties one wants to examine (see [4,7] or the more recent [12]). The semigroup tree is related to the function on numerical semigroups defined by S → S ∪ {F(S)}. ...

Reference: On some numerical semigroup transforms

... " to indicate that the semigroup consecutively contains all the integers from the number that precedes the ellipsis. The semigroup H arises in the mathematical theory of music [3]. It is obviously cofinite and it contains zero. ...

... Hoholdt et al. [8] provided a survey of the existing literature on the decoding of algebraic geometric codes. To study different concepts related to numerical semigroups and their applications in coding theory, the readers can see [9]. Let N be a set of nonnegative integers. ...