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In this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

We prove a general structure theorem by which an inside factorial monoid can be uniquely decomposed into a factorial monoid and the related Apery set. The structure of the latter is analyzed subsequently. As an example, we study the Cale monoid of nonnegative solutions of equations of the form a 1 x 1 + ⋯ + a r − 1 x r − 1 = a r x r for positive integers a 1 , … , a r . In this case, the monoid is isomorphic to a simplicial full affine semigroup, and the factorial monoid of the decomposition is generated by extremal rays. For the special case r = 3 , we discuss the relationship to a kind of unique representation as studied by Elliott for a r ≤ 10 .

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these resutls, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

This work extends the results known for the Delta sets of non-symmetric numerical semigroups with embedding dimension three to the symmetric case. Thus, we have a fast algorithm to compute the Delta set of any embedding dimension three numerical semigroup. Also, as a consequence of these results, the sets that can be realized as Delta sets of numerical semigroups of embedding dimension three are fully characterized.

As far as we know, usual computer algebra packages can not compute denumerants for almost medium (about a hundred digits) or almost medium--large (about a thousand digits) input data in a reasonably time cost on an ordinary computer. Implemented algorithms can manage numerical $n$--semigroups for small input data. Here we are interested in denumerants of numerical $3$--semigroups which have almost medium input data. A new algorithm for computing denumerants is given for this task. It can manage almost medium input data in the worst case and medium--large or even large input data in some cases., Comment: 32 pages, 5 figures, paper

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of �( S) and the Betti elements of S. We prove how the minimum and maximum element of �( S) can be determined using the Betti elements of S. This leads to a determination of when �( S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.

Questions concerning the lengths of factorizations into irreducible elements in numerical monoids have gained much attention in the recent literature. In this note, we show that a numerical monoid has an element with two different irreducible factorizations of the same length if and only if its embedding dimension is greater than two. We find formulas in embedding dimension three for the smallest element with two different irreducible factorizations of the same length and the largest element whose different irreducible factorizations all have distinct lengths. We show that these formulas do not naturally extend to higher embedding dimensions.

Minimum distance diagrams, also known as L-shapes, have been used to study some properties related to weighted Cayley digraphs of degree two and embedding dimension three numerical semigroups. In this particular case, it has been shown that these discrete structures have at most two related L-shapes. These diagrams are proved to be a good tool for studying factorizations and the catenary degree for semigroups and diameter and distance between vertices for digraphs.; This maximum number of L-shapes has not been proved to be kept when increasing the degree of digraphs or the embedding dimension of semigroups. In this work we give a family of embedding dimension four numerical semigroups S-n, for odd n >= 5, such that the number of related L-shapes is n+3/2. This family has her analogue to weighted Cayley digraphs of degree three.; Therefore, the number of L-shapes related to numerical semigroups can be as large as wanted when the embedding dimension is at least four. The same is true for weighted Cayley digraphs of degree at least three. This fact has several implications on the combinatorics of factorizations for numerical semigroups and minimum paths between vertices for weighted digraphs. (C) 2015 Elsevier B.V. All rights reserved.

Problems involving chains of irreducible factorizations in atomic integral domains and monoids have been the focus of much recent literature. If S is a commutative cancellative atomic monoid, then the catenary degree of S (denoted c(S)) and the tame degree of S (denoted t(S)) are combinatorial invariants of S which describe the behavior of chains of factorizations. In this note, we describe methods to compute both c(S) and t(S) when M is a finitely generated commutative cancellative monoid.

We define formally smooth coalgebras and we study their relation with hereditary coalgebras. The main result of the paper establishes that a coalgebra with separable coradical is hereditary if and only if it is formally smooth if and only if it is the cotensor coalgebra \(T_{C_{0}}(N)\) , where C0 is the coradical of C and N is a certain (C0,C0)–bicomodule.

The aim of this paper is to avoid some difficulties, related with the Lie bracket, in the definition of vector fields in a noncommutative setting, as they were defined by Woronowicz, Schmudgen-Schuler and Aschieri-Schupp. We extend the definition of vector fields to consider them as “derivations” of the algebra, through Cartan pairs introduced by Borowiec. Then, using translations, we introduce the invariant vector fields. Finally, the definition of Lie bracket realized by Dubois-Violette, considering elements in the center of the algebra, is also extended to these invariant vector fields.

We give an algorithm to compute the set of primitive elements for an embedding dimension three numerical semigroups. We show how we use this procedure in the study of the construction of L-shapes and the tame degree of the semigroup.

Denumerants of numerical semigroups are known to be difficult to obtain, even with small embedding dimension of the semigroups. In this work we give some results on denumerants of 3-semigroups S = 〈 a , b , c 〉 . Closed expressions are obtained under certain conditions.

The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behaviour of such bounds for r > 1 differs from that of the classical Feng-Rao distance (r=1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Ap��ry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower-bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases., 20 pages, 14 figures

We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the $r^{th}$ Feng-Rao number is obtained., 23 pages, 6 figures

Varying methods exist for computing a presentation of a finitely generated commutative cancellative monoid. We use an algorithm of Contejean and Devie [An efficient incremental algorithm for solving systems of linear diophantine equations, Inform. and Comput. 113 (1994) 143–172] to show how these presentations can be obtained from the nonnegative integer solutions to a linear system of equations. We later introduce an alternate algorithm to show how such a presentation can be efficiently computed from an integer basis.