Semigroup Algebras Research Articles

Minimal free resolution of generalized repunit algebras

Let k be an arbitrary field and let b > 1 , n > 1 and a be three positive integers. In this paper we explicitly describe a minimal S-graded free resolution of the semigroup algebra k [ S ] when S is a generalized repunit numerical semigroup, that is, when S is the submonoid of N generated by { a 1 , a 2 , … , a n } where a 1 = ∑ j = 0 n − 1 b j and a i − a i − 1 = a b i − 2 , i = 2 , … , n , with gcd ( a , a 1 ) = 1 .

Exel–Pardo algebras with a twist

Takeshi Katsura associated a C^{\ast} -algebra C^{\ast}_{A,B} to a pair of square matrices A\ge 0 and B of the same size with integral coefficients and gave sufficient conditions on (A,B) to be simple purely infinite (SPI). We call such a pair a KSPI pair. It follows from a result of Katsura that any separable C^{\ast} -algebra \mathfrak{A} which is a cone of a map \xi:C(S^{1})^{n}\rightarrow C(S^{1})^{n} in Kasparov’s triangulated category KK is KK -isomorphic to C^{\ast}_{A,B} for some KSPI pair (A,B) . In this article, we introduce, for the data of a commutative ring \ell , non-necessarily square matrices A , B and a matrix C of the same size with coefficients in the group \mathcal{U}(\ell) of invertible elements, an \ell -algebra \mathcal{O}_{A,B}^{C} , the twisted Katsura algebra of the triple (A,B,C) . When A and B are square and C is trivial, we recover the Katsura \ell -algebra first considered by Enrique Pardo and Ruy Exel. We show that if \ell is a field of characteristic 0 and (A,B) is KSPI, then \mathcal{O}_{A,B}^{C} is SPI, and that any \ell -algebra which is a cone of a map \xi:\ell[t,t^{-1}]^{n}\rightarrow \ell[t,t^{-1}]^{n} in the triangulated bivariant algebraic K -theory category kk is kk -isomorphic to \mathcal{O}_{A,B}^{C} for some (A,B,C) as above so that (A,B) is KSPI. Katsura \ell -algebras are examples of the Exel–Pardo algebras L(G,E,\phi) associated to a group G acting on a directed graph E and a 1 -cocycle \phi:G\times E^{1}\rightarrow G . Similarly, twisted Katsura algebras are examples of the twisted Exel–Pardo \ell -algebras L(G,E,\phi_{c}) we introduce in the current article; they are associated to data (G,E,\phi) as above twisted by a 1 -cocycle c:G\times E^{1}\rightarrow \mathcal{U}(\ell) . The algebra L(G,E,\phi_{c}) can be variously described by generators and relations, as a quotient of a twisted semigroup algebra, as a twisted Steinberg algebra, as a corner skew Laurent polynomial algebra, and as a universal localization of a tensor algebra. We use each of these guises of L(G,E,\phi_{c}) to study its K -theoretic, regularity, and (purely infinite) simplicity properties. For example, we show that if \ell is a field of characteristic 0 , G and E are countable, and E is regular, then L(G,E,\phi_{c}) is simple whenever the Exel–Pardo C^{\ast} -algebra C^{\ast}(G,E,\phi) is, and is SPI if in addition the Leavitt path algebra L(E) is SPI.

Identities of Hecke–Kiselman monoids

AbstractIt is shown that the Hecke–Kiselman monoid $${\text {HK}}_{\Theta }$$ HK Θ associated to a finite oriented graph $$\Theta $$ Θ satisfies a semigroup identity if and only if $${\text {HK}}_{\Theta }$$ HK Θ does not have free noncommutative subsemigroups. It follows that this happens exactly when the semigroup algebra $$K[{\text {HK}}_{\Theta }]$$ K [ HK Θ ] over a field K satisfies a polynomial identity. The latter is equivalent to a condition expressed in terms of the graph $$\Theta $$ Θ . The proof allows to derive concrete identities satisfied by such monoids $${\text {HK}}_{\Theta }$$ HK Θ .

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

The determinant of semigroups of the pseudovariety ECOM

The purpose of this paper is to compute the nonzero semigroup determinant of the class of finite semigroups in which every two idempotents commute. This class strictly contains the class of finite semigroups that have central idempotents and the class of finite inverse semigroups. This computation holds significance in the context of the extension of the MacWilliams theorem for codes over semigroup algebras.

SEMI GROUP IDENTITIES WITH APPLICATIONS TO SEMI GROUPS

This research paper explores the rich and intriguing world of semi-group identities, their properties, and their applications to various types of semi-groups. Semi-groups are algebraic structures that generalize the notion of groups, allowing for non-invertible elements. Despite their broader scope, semi-groups retain many important features found in group theory. This study investigates the identities that hold true in the context of semi-group algebra and sheds light on the underlying mathematical structures and relationships among these identities. By delving into specific applications, we illustrate the significance of these findings to various types of semi-groups, such as monoids, semigroups with zero, and cancellative semi-groups. Ultimately, our results not only deepen our understanding of the fundamental properties of semi-groups. But also provide valuable insights for researchers in the areas of algebraic structures, combinatorics, and theoretical computer science.

On amenable-like properties of some Banach algebras with applications to study their nilpotent ideals

In this current article, we show that a non-zero closed nilpotent ideal in a Johnson pseudo-contractible Banach algebra cannot be [Formula: see text]-predual. We prove also that a nilpotent ideal in a Johnson pseudo-contractible Banach algebras with the approximation property is forced to be zero. Among other things, we characterize the property [Formula: see text] of some semigroup algebras associated with inverse semigroups. More especially, for an inverse semigroup [Formula: see text] such that [Formula: see text] is uniformly locally finite, [Formula: see text] has the property [Formula: see text] if and only if each maximal subgroup of [Formula: see text] is amenable, where [Formula: see text] is the set of idempotents of [Formula: see text]. Furthermore, we study the property [Formula: see text] of some [Formula: see text]-Lau product structures.

Algebras of reduced E-Fountain semigroups and the generalized ample identity II

We study the generalized right ample identity, introduced by the author in a previous paper. Let [Formula: see text] be a reduced [Formula: see text]-Fountain semigroup which satisfies the congruence condition. We can associate with [Formula: see text] a small category [Formula: see text] whose set of objects is identified with the set [Formula: see text] of idempotents and its morphisms correspond to elements of [Formula: see text]. We prove that [Formula: see text] satisfies the generalized right ample identity if and only if every element of [Formula: see text] induces a homomorphism of left [Formula: see text]-actions between certain classes of generalized Green’s relations. In this case, we interpret the associated category [Formula: see text] as a discrete form of a Peirce decomposition of the semigroup algebra. We also give some natural examples of semigroups satisfying this identity.

Some Characterizations for Approximate Biflatness of Semigroup Algebras

In this paper, we study an approximate biflatness ofl1S, whereSis a Clifford semigroup. Indeed, we show that a Clifford semigroup algebral1Sis approximately biflat if and only if every maximal subgroup ofSis amenable,ESis locally finite, andl1Shas an approximate identity inc00S. Moreover, we prove thatl1Sis approximately biflat if and only if each maximal subgroup ofSis amenable for an inverse semigroupSsuch thatES, the set of its idempotent elements, is totally ordered and locally finite.

On the associated graded ring of semigroup algebras

In this paper, we give a necessary and sufficient condition for the Cohen-Macaulayness of the associated graded ring of a simplicial affine semigroup using Gröbner basis. We generalize the concept of homogeneous numerical semigroup for the simplicial affine semigroup and show that the Betti numbers of the corresponding semigroup ring matches with the Betti numbers of the associated graded ring. We also discuss nice extensions for simplicial affine semigroups, a concept which is motivated by nice extensions of numerical semigroups.

Finite-Dimensional Approximations and Semigroup Coactions for Operator Algebras

Abstract The residual finite-dimensionality of a ${\textrm{C}}^{\ast }$-algebra is known to be encoded in a topological property of its space of representations, stating that finite-dimensional representations should be dense therein. We extend this paradigm to general (possibly non-self-adjoint) operator algebras. While numerous subtleties emerge in this greater generality, we exhibit novel tools for constructing finite-dimensional approximations. One such tool is a notion of a residually finite-dimensional coaction of a semigroup on an operator algebra, which allows us to construct finite-dimensional approximations for operator algebras of functions and operator algebras of semigroups. Our investigation is intimately related to the question of when residual finite-dimensionality of an operator algebra is inherited by its maximal ${\textrm{C}}^{\ast }$-cover, which we establish in many cases of interest.

On the Automorphism Group of Non-Necessarily Normal Affine Toric Varieties

Abstract Our main result is the following: let $X$ be a normal affine toric surface without torus factor. Then there exists a non-normal affine toric surface $X^{\prime}$ with automorphism group isomorphic to the automorphism group of $X$ if and only if $X$ is different from the affine plane. As a tool, we first provide a classification of normalized additive group actions on a non-necessarily normal affine toric variety $X$ of any dimension. Recall that normalized additive group actions on $X$ are in correspondence with homogeneous locally nilpotent derivations on the algebra of regular functions of $X$. More generally, we provide a classification of homogeneous locally nilpotent derivations on the semigroup algebra of a commutative cancellative monoid.

On noncommutative bounded factorization domains and prime rings

A ring has bounded factorizations if every cancellative nonunit a∈R can be written as a product of atoms and there is a bound λ(a) on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations.

On K-theoretic invariants of semigroup C*-algebras from actions of congruence monoids

We study semigroup $C^*$-algebras of semigroups associated with number fields and initial data arising naturally from class field theory. These semigroup $C^*$-algebras turn out to have an interesting $C^*$-algebraic structure, giving access to many new examplesof classifiable $C^*$-algebras and exhibi ting phenomena which have not appeared before. Moreover, using K-theoretic invariants, we investigate how much information about the initial number-theoretic data is encoded in our semigroup $C^*$-algebras.

Semigroup algebras which are Azumaya algebras

Semigroup algebras which are Azumaya algebras

On approximately biprojective and approximately biflat Banach algebras

In this paper, we study the approximate biprojectivity and the approximate biflatness of a Banach algebra A and find some relations between theses concepts with ?-amenability and ? -contractibility, where ? is a character on A. Among other things, we show that ?-Lau product algebra L1(G) ?? A(G) is approximately biprojective if and only if G is finite, where L1(G) and A(G) are the group algebra and the Fourier algebra of a locally compact group G, respectively. We also characterize approximately biprojective and approximately biflat semigroup algebras associated with the inverse semigroups.

Centrally essential semigroup algebras

Centrally essential semigroup algebras

First module cohomology group of induced semigroup algebras

‎Let $S$ be a discrete semigroup and $T$ be a left multiplier operator on $S$‎. ‎A new product on $S$ defined by $T$ creates a new induced semigroup $S _{T} $‎. ‎In this paper‎, ‎we show that if $T$ is bijective‎, ‎then the first module cohomology groups $ \HH_{\ell^1(E)}^{1}(\ell^1(S)‎, ‎\ell^{\infty}(S))$ and $ \HH_{\ell^1(E_{T})}^{1}(\ell^1({S_{T}})‎, ‎\ell^{\infty}(S_{T})) $ are equal‎, ‎where $E$ and $E_{T}$ are sets of idempotent elements in $S$ and $S _{T}$‎, ‎respectively‎. ‎Which in particular means that $\ell^1(S)$ is weak $\ell^1(E)$-module amenable if and only if $\ell^1(S_T)$ is weak $\ell^1(E_T)$-module amenable‎. ‎Finally‎, ‎by giving an example‎, ‎we show that the condition of bijectivity for $T$‎, ‎is necessary‎.

Centrally essential semigroup algebras

For a cancellative semigroup S and a field F, it is proved that the semigroup algebra FS is centrally essential if and only if the group of fractions GS of the semigroup S exists and the group algebra FGS of GS is centrally essential. The semigroup algebra of a cancellative semigroup is centrally essential if and only if it has a classical right ring of fractions which is a centrally essential ring. There exist non-commutative centrally essential semigroup algebras over fields of zero characteristic (this contrasts with the known fact that centrally essential group algebras over fields of zero characteristic are commutative).

On atomic density of numerical semigroup algebras

A numerical semigroup S is a cofinite, additively closed subset of the nonnegative integers that contains 0. We initiate the study of atomic density, an asymptotic measure of the proportion of irreducible elements in a given ring or semigroup, for semigroup algebras. It is known that the atomic density of the polynomial ring 𝔽q[x] is zero for any finite field 𝔽q; we prove that the numerical semigroup algebra 𝔽q[S] also has atomic density zero for any numerical semigroup S. We also examine the particular algebra 𝔽2[x2,x3] in more detail, providing a bound on the rate of convergence of the atomic density as well as a counting formula for irreducible polynomials using Möbius inversion, comparable to the formula for irreducible polynomials over a finite field 𝔽q.