Related papers: Monoidal networks
Let $C \subset {\bf N}^d$ be an affine semigroup, and $R=K[C]$ its semigroup ring. This paper is a collection of various results on "$C$-graded" $R$-modules, especially, monomial ideals. For example, we show the following: If $R$ is normal…
This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…
In this thesis, we present a flexible framework for specifying and constructing operads which are suited to reasoning about network construction. The data used to present these operads is called a \emph{network model}, a monoidal variant of…
Irreducible decompositions of monomial ideals in polynomial rings over a field are well-understood. In this paper, we investigate decompositions in the set of monomial ideals in the semigroup ring A[\mathbb{R}_{\geq 0}^d] where A is an…
We define the Hochschild complex and cohomology of a ring object in a monoidal category enriched over abelian groups. We interpret the cohomology groups and prove that the cohomology ring is graded-commutative.
In this article, we introduce a generalization of the concept of graded $r$-ideals in graded commutative rings with nonzero unity. Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all…
Given an ideal I in a polynomial ring, we consider the largest monomial subideal contained in I, denoted mono(I). We study mono as an interesting operation in its own right, guided by questions that arise from comparing the Betti tables of…
In [arXiv:1509.02937], the notion of a module tensor category was introduced as a braided monoidal central functor $F\colon \mathcal{V}\longrightarrow \mathcal{T}$ from a braided monoidal category $\mathcal{V}$ to a monoidal category…
The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a…
Let R be a commutative ring with unity, M be an unitary R-module and {\Gamma} be a simple graph. This research article is an interplay of combinatorial and algebraic properties of M . We show a combinatorial object completely determines an…
Monoidal width was recently introduced by the authors as a measure of the complexity of decomposing morphisms in monoidal categories. We have shown that in a monoidal category of cospans of graphs, monoidal width and its variants can be…
In this article, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides natural and canonical approaches in order to find easy and rigorous proofs and methods for…
In this paper, we consider graded near-rings over a monoid $G$ as a generalizations of graded rings over groups. We introduce certain innovative graded prime ideals and study some of its basic properties over graded near-rings.
In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative…
Let M be a monoidal category endowed with a distinguished class of weak equivalences and with appropriately compatible classifying bundles for monoids and comonoids. We define and study homotopy-invariant notions of normality for maps of…
We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly,…
Let $R$ be a commutative ring with identity. We define a graph $\Gamma_{\aut}(R)$ on $ R$, with vertices elements of $R$, such that any two distinct vertices $x, y$ are adjacent if and only if there exists $\sigma \in \aut$ such that…
Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize…
Let R be a commutative ring with identity, and let I be an ideal of R. The zero-divisor graph of R with respect to I, denoted by $\Gamma_I(R)$, is the graph whose vertices are the set $\{x \in R \setminus I | xy \in I$ for some $y \in R…