Related papers: Sorting and labelling integral ideals in a number …

We consider the problem of sorting $n$ items, given the outcomes of $m$ pre-existing comparisons. We present a simple and natural deterministic algorithm that runs in $O(m + \log T)$ time and does $O(\log T)$ comparisons, where $T$ is the…

Let $\mathcal{A}$ denote a central hyperplane arrangement of rank $n$ in affine space $\mathbb{K}^n$ over an infinite field $\mathbb{K}$ and let $l_1,\ldots, l_m\in R:= \mathbb K[x_1,\ldots,x_n]$ denote the linear forms defining the…

We establish a series of results showing that the Jacobian ideal is contained in the test ideal. We first prove a new result in characteristic $p$ for complete rings over a field $K$. Then we prove some results showing that Jacobian ideals…

Suppose we are given black-box access to a finite ring R, and a list of generators for an ideal I in R. We show how to find an additive basis representation for I in poly(log |R|) time. This generalizes a quantum algorithm of Arvind et al.…

In this paper, a sorting technique is presented that takes as input a data set whose primary key domain is known to the sorting algorithm, and works with an time efficiency of O(n+k), where k is the primary key domain. It is shown that the…

There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes…

Learning rules plays a crucial role in deep learning, particularly in explainable artificial intelligence and enhancing the reasoning capabilities of large language models. While existing rule learning methods are primarily designed for…

Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…

We present an alternative method for computing primary decomposition of zero-dimensional ideals over finite fields. Based upon the further decomposition of the invariant subspace of the Frobenius map acting on the quotient algebra in the…

We introduce the notion of sortability and $t$-sortability for a simplicial complex and study the graphs for which their independence complexes are either sortable or $t$-sortable. We show that the proper interval graphs are precisely the…

Taking a ring-theoretic perspective as our motivation, the main aim of this series is to establish a comprehensive theory of ideals in commutative quantales with an identity element. This particular article focuses on an examination of…

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the…

The entities in directed networks arising from real-world interactions are often naturally organized under some hierarchical structure. Given a directed, weighted, graph with edges and node labels, we introduce ranking problem where the…

Let S be a polynomial ring in n variables, over an arbitrary field. We give the total, graded, and multigraded Betti numbers of S/M, for every monomial ideal M in S. We also give an explicit characterization of all monomial ideals M in S…

We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in…

In this paper it is shown that a sortable ideal $I$ is Freiman if and only if its sorted graph is chordal. This characterization is used to give a complete classification of Freiman principal Borel ideals and of Freiman Veronese type ideals…

Extreme multi-label learning (XML) or classification has been a practical and important problem since the boom of big data. The main challenge lies in the exponential label space which involves $2^L$ possible label sets especially when the…

Graph labeling is a technique that assigns unique labels or weights to the vertices or edges of a graph, often used to analyze and solve various graph-related problems. There are few methods with certain limitations conducted by researchers…

Multi-label learning is concerned with the classification of data with multiple class labels. This is in contrast to the traditional classification problem where every data instance has a single label. Due to the exponential size of output…

Let $S_n$ be a polynomial ring with $n$ variables over a field and $\{I_n\}_{n \geq 1}$ a chain of ideals such that each $I_n$ is a monomial ideal of $S_n$ fixed by permutations of the variables. In this paper, we present a way to determine…