Related papers: Quotients of numerical semigroups generated by two…
This work delves into the {\it quotient of an affine semigroup by a positive integer}, exploring its intricate properties and broader implications. We unveil an {\it associated tree} that serves as a valuable tool for further analysis.…
Let a, b be non-zero complex numbers and l an odd natural number bigger that 2. We determine all Hopf algebra quotients of the quantized coordinate algebra O_{a,b}(GL_{n}) when a^{-1}b is a primitive l-th root of unity and a, b satisfy…
We study Wilf's conjecture for numerical semigroups $S$ such that the second least generator $a_2$ of $S$ satisfies $a_2>\frac{c(S)+\mu(S)}{3}$, where $c(S)$ is the conductor and $\mu(S)$ the multiplicity of $S$. In particular, we show that…
We introduce the notion of numerical semigroups generated by concatenation of arithmetic sequences and show that this class of numerical semigroups exhibit multiple interesting behaviours.
Let $\langle A\rangle$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle…
Let $\mathsf{r}_k$ be the unique positive root of $x^k - (x+1)^{k-1} = 0$. We prove the best known bounds on the number $n_{g,d}$ of $d$-dimensional generalized numerical semigroups, in particular that \[n_{g,d} > C_d^{g^{(d-1)/d}}…
A quasi-automatic semigroup is defined by a finite set of generators, a rational (regular) set of representatives, such that if a is a generator or neutral, then the graph of right multiplication by a on the set of representatives is a…
We study numerical semigroups with the property "multiplicity= embedding dimension+1", generated by concatenation of arithmetic sequences.
For a given inverse semigroup, one can associate an \'etale groupoid which is called the universal groupoid. Our motivation is studying the relation between inverse semigroups and associated \'etale groupoids. In this paper, we focus on…
An explicit form of the generators of quantum and ordinary semisimple algebras for an arbitrary finite-dimensional representation is found. The generators corresponding to the simple roots are obtained in terms of a solution of a system of…
Let $\mathbb{F}$ be a finite field of odd order and $a,b\in\mathbb{F}\setminus\{0,1\}$ be such that $\chi(a) = \chi(b)$ and $\chi(1-a)=\chi(1-b)$, where $\chi$ is the extended quadratic character. Let $Q_{a,b}$ be the quasigroup upon…
Let $a,b$ be positive integers. In this note, we study the numerical semigroup $H=\left<a,a+1,b\right>$ and and the associated numerical semigroup ring $R=k[[H]]$. Under the certain conditions, we provide explicit formulas for the Frobenius…
In this paper we introduce the notion of extension of a numerical semigroup. We provide a characterization of the numerical semigroups whose extensions are all arithmetic and we give an algorithm for the computation of the whole set of…
In this work we present a new class of numerical semigroups called GSI-semigroups. We see the relations between them and others families of semigroups and we give explicitly their set of gaps. Moreover, an algorithm to obtain all the…
We define the concentration of a numerical semigroup $S$ as $\mathsf{C}(S)=\max \left\{\text{next}_S(s)-s ~|~ s\in S \backslash \{0\}\right\}$ wherein $\text{next}_S(s)=\min\left\{x \in S ~|~ s<x\right\}$. In this paper, we study the class…
Let $a$ and $d$ be two linearly independent vectors in $\mathbb{N}^2$, over the field of rational numbers. For a positive integer $k \geq 2$, consider the sequence $a, a+d, \ldots, a+kd$ such that the affine semigroup $S_{a,d,k} = \langle…
Numerical semigroups have been extensively studied throughout the literature, and many of their invariants have been characterized. In this work, we generalize some of the most important results about symmetry, pseudo-symmetry, or…
In this paper, we extend recent results about the distribution of even and odd gaps of a numerical semigroup. We find that, for any numerical semigroup, the distribution can be computed in terms of the numbers of or the sums of odd and even…
The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$, and finding the Frobenius number is called the Frobenius problem. In this paper, we solve the Frobenius problem for the numerical…
Some varieties of groupoids and quasigroups generated by linear-bivariate polynomials $P(x,y)=a+bx+cy$ over the ring $\mathbb{Z}_n$ are studied. Necessary and sufficient conditions for such groupoids and quasigroups to obey identities which…