Certain partitions on a set and their applications to different classes of graded algebras

Antonio J. Calderón Martín; Boubacar Dieme

Displaying similar documents to “Certain partitions on a set and their applications to different classes of graded algebras”

Commutative graded- $S$ -coherent rings

Anass Assarrar, Najib Mahdou, Ünsal Tekir, Suat Koç (2023)

Czechoslovak Mathematical Journal

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Recently, motivated by Anderson, Dumitrescu’s $S$ -finiteness, D. Bennis, M. El Hajoui (2018) introduced the notion of $S$ -coherent rings, which is the $S$ -version of coherent rings. Let $R = ⨁_{α \in G} R_{α}$ be a commutative ring with unity graded by an arbitrary commutative monoid $G$ , and $S$ a multiplicatively closed subset of nonzero homogeneous elements of $R$ . We define $R$ to be graded- $S$ -coherent ring if every finitely generated homogeneous ideal of $R$ is $S$ -finitely presented. The purpose of this paper is to give...

$G r$ - $(2, n)$ -ideals in graded commutative rings

Khaldoun Al-Zoubi, Shatha Alghueiri, Ece Y. Celikel (2020)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a group with identity $e$ and let $R$ be a $G$ -graded ring. In this paper, we introduce and study the concept of graded $(2, n)$ -ideals of $R$ . A proper graded ideal $I$ of $R$ is called a graded $(2, n)$ -ideal of $R$ if whenever $r s t \in I$ where $r, s, t \in h (R)$ , then either $r t \in I$ or $r s \in G r (0)$ or $s t \in G r (0)$ . We introduce several results concerning $g r$ - $(2, n)$ -ideals. For example, we give a characterization of graded $(2, n)$ -ideals and their homogeneous components. Also, the relations between graded $(2, n)$ -ideals and others that already exist, namely, the graded prime...

The $G$ -graded identities of the Grassmann Algebra

Lucio Centrone (2016)

Archivum Mathematicum

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Let $G$ be a finite abelian group with identity element $1_{G}$ and $L = ⨁_{g \in G} L^{g}$ be an infinite dimensional $G$ -homogeneous vector space over a field of characteristic $0$ . Let $E = E (L)$ be the Grassmann algebra generated by $L$ . It follows that $E$ is a $G$ -graded algebra. Let $| G |$ be odd, then we prove that in order to describe any ideal of $G$ -graded identities of $E$ it is sufficient to deal with $G^{'}$ -grading, where $| G^{'} | \leq | G |$ , ${dim}_{F} L^{1_{G^{'}}} = \infty$ and ${dim}_{F} L^{g^{'}} < \infty$ if $g^{'} \neq 1_{G^{'}}$ . In the same spirit of the case $| G |$ odd, if $| G |$ is even it is sufficient to study only those $G$ -gradings...

$n$ - $gr$ -coherent rings and Gorenstein graded modules

Mostafa Amini, Driss Bennis, Soumia Mamdouhi (2022)

Czechoslovak Mathematical Journal

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Let $R$ be a graded ring and $n \geq 1$ be an integer. We introduce and study the notions of Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules by using the notion of special finitely presented graded modules. On $n$ -gr-coherent rings, we investigate the relationships between Gorenstein $n$ -FP-gr-injective and Gorenstein $n$ -gr-flat modules. Among other results, we prove that any graded module in $R$ -gr (or gr- $R$ ) admits a Gorenstein $n$ -FP-gr-injective (or Gorenstein $n$ -gr-flat) cover and preenvelope,...

On the symmetric algebra of certain first syzygy modules

Gaetana Restuccia, Zhongming Tang, Rosanna Utano (2022)

Czechoslovak Mathematical Journal

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Let $(R, 𝔪)$ be a standard graded $K$ -algebra over a field $K$ . Then $R$ can be written as $S / I$ , where $I \subseteq {(x_{1}, ..., x_{n})}^{2}$ is a graded ideal of a polynomial ring $S = K [x_{1}, ..., x_{n}]$ . Assume that $n \geq 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${Sym}_{R} ({Syz}_{1} (𝔪))$ of the first syzygy module ${Syz}_{1} (𝔪)$ of $𝔪$ . When the minimal generators of $I$ are all of degree 2, the dimension of ${Sym}_{R} ({Syz}_{1} (𝔪))$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.

Singularity categories of skewed-gentle algebras

Xinhong Chen, Ming Lu (2015)

Colloquium Mathematicae

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Let K be an algebraically closed field. Let (Q,Sp,I) be a skewed-gentle triple, and let $(Q^{s g}, I^{s g})$ and $(Q^{g}, I^{g})$ be the corresponding skewed-gentle pair and the associated gentle pair, respectively. We prove that the skewed-gentle algebra $K Q^{s g} / ⟨ I^{s g} ⟩$ is singularity equivalent to KQ/⟨I⟩. Moreover, we use (Q,Sp,I) to describe the singularity category of $K Q^{g} / ⟨ I^{g} ⟩$ . As a corollary, we find that $g l d i m K Q^{s g} / ⟨ I^{s g} ⟩ < \infty$ if and only if $g l d i m K Q / ⟨ I ⟩ < \infty$ if and only if $g l d i m K Q^{g} / ⟨ I^{g} ⟩ < \infty$ .

Betti numbers of some circulant graphs

Mohsen Abdi Makvand, Amir Mousivand (2019)

Czechoslovak Mathematical Journal

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Let $o (n)$ be the greatest odd integer less than or equal to $n$ . In this paper we provide explicit formulae to compute $ℕ$ -graded Betti numbers of the circulant graphs $C_{2 n} (1, 2, 3, 5, ..., o (n))$ . We do this by showing that this graph is the product (or join) of the cycle $C_{n}$ by itself, and computing Betti numbers of $C_{n} * C_{n}$ . We also discuss whether such a graph (more generally, $G * H$ ) is well-covered, Cohen-Macaulay, sequentially Cohen-Macaulay, Buchsbaum, or $S_{2}$ .

Quantised ${𝔰𝔩}_{2}$ -differential algebras

Andrey Krutov, Pavle Pandžić (2024)

Archivum Mathematicum

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We propose a definition of a quantised ${𝔰𝔩}_{2}$ -differential algebra and show that the quantised exterior algebra (defined by Berenstein and Zwicknagl) and the quantised Clifford algebra (defined by the authors) of ${𝔰𝔩}_{2}$ are natural examples of such algebras.

Stratified modules over an extension algebra

Erzsébet Lukács, András Magyar (2018)

Czechoslovak Mathematical Journal

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Let $A$ be a standard Koszul standardly stratified algebra and $X$ an $A$ -module. The paper investigates conditions which imply that the module ${Ext}_{A}^{*} (X)$ over the Yoneda extension algebra $A^{*}$ is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of $A$ is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.

Trivialization of $𝒞 (X)$ -algebras with strongly self-absorbing fibres

Marius Dadarlat, Wilhelm Winter (2008)

Bulletin de la Société Mathématique de France

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Suppose $A$ is a separable unital $𝒞 (X)$ -algebra each fibre of which is isomorphic to the same strongly self-absorbing and $K_{1}$ -injective $C^{*}$ -algebra $𝒟$ . We show that $A$ and $𝒞 (X) \otimes 𝒟$ are isomorphic as $𝒞 (X)$ -algebras provided the compact Hausdorff space $X$ is finite-dimensional. This statement is known not to extend to the infinite-dimensional case.

The bicrossed products of $H_{4}$ and $H_{8}$

Daowei Lu, Yan Ning, Dingguo Wang (2020)

Czechoslovak Mathematical Journal

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Let $H_{4}$ and $H_{8}$ be the Sweedler’s and Kac-Paljutkin Hopf algebras, respectively. We prove that any Hopf algebra which factorizes through $H_{8}$ and $H_{4}$ (equivalently, any bicrossed product between the Hopf algebras $H_{8}$ and $H_{4}$ ) must be isomorphic to one of the following four Hopf algebras: $H_{8} \otimes H_{4}, H_{32, 1}, H_{32, 2}, H_{32, 3}$ . The set of all matched pairs $(H_{8}, H_{4}, ▹, ◃)$ is explicitly described, and then the associated bicrossed product is given by generators and relations.

Symmetries in connected graded algebras and their PBW-deformations

Yongjun Xu, Xin Zhang (2023)

Czechoslovak Mathematical Journal

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We focus on connected graded algebras and their PBW-deformations endowed with additional symmetric structures. Many well-known algebras such as negative parts of Drinfeld-Jimbo’s quantum groups, cubic Artin-Schelter algebras and three-dimensional Sklyanin algebras appear in our research framework. As an application, we investigate a $𝒦_{2}$ algebra $𝒜$ which was introduced to compute the cohomology ring of the Fomin-Kirillov algebra ${ℱ𝒦}_{3}$ , and explicitly construct all the (self-)symmetric and sign-(self-)symmetric...

Structure fractals and para-quaternionic geometry

Julian Ławrynowicz, Massimo Vaccaro (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions $p = 2, 4$ and $8$ , respectively, but the procedure fails for $p = 16$ in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are $n = 2^{7}$ . Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower...

Transfer of derived equivalences from subalgebras to endomorphism algebras II

Shengyong Pan, Jiahui Yu (2024)

Czechoslovak Mathematical Journal

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We investigate derived equivalences between subalgebras of some $Φ$ -Auslander-Yoneda algebras from a class of $n$ -angles in weakly $n$ -angulated categories. The derived equivalences are obtained by transferring subalgebras induced by $n$ -angles to endomorphism algebras induced by approximation sequences. Then we extend our constructions in T. Brüstle, S. Y. Pan (2016) to $n$ -angle cases. Finally, we give an explicit example to illustrate our result.

A generalization of a formalized theory of fields of sets on non-classical logics

Helena Rasiowa

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Contents Introduction.................................................................................................................................................. 3 § 1. System $𝒮$ of a propositional calculus...................................................................... 4 § 2. System $𝒮 *$ ..................................................................................................................... 5 § 3. $𝒮 *$ -algebras.....................................................................................................................

The moduli space of commutative algebras of finite rank

Bjorn Poonen (2008)

Journal of the European Mathematical Society

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The moduli space of rank- $n$ commutative algebras equipped with an ordered basis is an affine scheme $𝔅_{n}$ of finite type over $ℤ$ , with geometrically connected fibers. It is smooth if and only if $n \leq 3$ . It is reducible if $n \geq 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$ ). The relative dimension of $𝔅_{n}$ is $\frac{2}{27} n^{3} + O (n^{8 / 3})$ . The subscheme parameterizing étale algebras is isomorphic to ${GL}_{n} / S_{n}$ , which is of dimension only $n^{2}$ . For $n \geq 8$ , there exist algebras that are not limits of étale algebras. The dimension...

Matrix representation of finite effect algebras

Grzegorz Bińczak, Joanna Kaleta, Andrzej Zembrzuski (2023)

Kybernetika

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In this paper we present representation of finite effect algebras by matrices. For each non-trivial finite effect algebra $E$ we construct set of matrices $M (E)$ in such a way that effect algebras $E_{1}$ and $E_{2}$ are isomorphic if and only if $M (E_{1}) = M (E_{2})$ . The paper also contains the full list of matrices representing all nontrivial finite effect algebras of cardinality at most $8$ .

Cambrian fans

Nathan Reading, David E. Speyer (2009)

Journal of the European Mathematical Society

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For a finite Coxeter group $W$ and a Coxeter element $c$ of $W$ ; the $c$ -Cambrian fan is a coarsening of the fan defined by the reflecting hyperplanes of $W$ . Its maximal cones are naturally indexed by the $c$ -sortable elements of $W$ . The main result of this paper is that the known bijection cl $_{c}$ between $c$ -sortable elements and $c$ -clusters induces a combinatorial isomorphism of fans. In particular, the $c$ -Cambrian fan is combinatorially isomorphic to the normal fan of the generalized associahedron for...