Finite Algebras Research Articles

Graphs of finite algebras: edges, and connectivity

Graphs of finite algebras: edges, and connectivity

STRICTLY n-FINITE VARIETIES OF HEYTING ALGEBRAS

Abstract For any $n<\omega $ we construct an infinite $(n+1)$ -generated Heyting algebra whose n-generated subalgebras are of cardinality $\leq m_n$ for some positive integer $m_n$ . From this we conclude that for every $n<\omega $ there exists a variety of Heyting algebras which contains an infinite $(n+1)$ -generated algebra, but which contains only finite n-generated algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [4].

Graphs of finite algebras: maximality, rectangularity, and decomposition

Graphs of finite algebras: maximality, rectangularity, and decomposition

Universal coacting Hopf algebra of a finite dimensional Lie-Yamaguti algebra

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.

Noncommutative Nest Hardy Spaces and Martingales

Abstract Let $(\mathcal{M},\tau )$ be a finite von Neumann algebra equipped with a normalized faithful trace and let $\mathbb{A}$ be a nest of order type $\mathbb{N}$. The nest algebra is defined by $H_{\infty }^{r}(\mathbb{A})=\{x\in \mathcal{M}:ex=exe,e\in \mathbb{A}\}$, and the related noncommutative Hardy space $H_{p}^{r}(\mathbb{A})$ is the completion of $H_{\infty }^{r}(\mathbb{A})$ in $L_{p}(\mathcal{M})$. In the present paper, we make a connection between $H_{p}^{r}(\mathbb{A})$ and certain noncommutative martingale Hardy space via showing that triangular truncation is a concrete martingale transform. We include several related results for noncommutative nest Hardy spaces in the spirit of recent development of noncommutative martingale theory.

Exceptional Periodicity and Magic Star algebras: Generalized roots, gradings, Hermitian Vinberg T-algebras and their derivations

We introduce countably infinite series of finite dimensional generalizations of the exceptional Lie algebras: in fact, each exceptional Lie algebra (but g2) is the first element of an infinite series of finite dimensional algebras, which we name Magic Star algebras. All these algebras (but the first elements of the infinite series) are not Lie algebras, but nevertheless they have remarkable similarities with many characterizing features of the exceptional Lie algebras; they also enjoy a kind of periodicity (inherited by Bott periodicity), which we name Exceptional Periodicity. We analyze the graded algebraic structures arising in a certain projection (named Magic Star projection) of the generalized root systems pertaining to Magic Star algebras, and we highlight the occurrence of a class of rank-3, Hermitian matrix (special Vinberg T)-algebras (which we call H algebras) on each vertex of such a projection. We then focus on the Magic Star algebra f4(n), which generalizes the non-simply laced exceptional Lie algebra f4, and deserves a treatment apart. Finally, we compute the Lie algebra of the inner derivations of the H algebras, pointing out the enhancements occurring for each first element of the series of Magic Star algebras, thus retrieving the result known for the derivations of cubic simple Jordan algebras.

Stone Pseudovarieties

We extend the theory of profinite algebras, as it is used in the algebraic theory of regular languages, to the more general setting of Stone topological algebras with the ultimate goal of going beyond classes of regular languages. We introduce Stone pseudovarieties, that is, classes of Stone topological algebras of a fixed topological signature that are closed under taking Stone quotients, closed subalgebras and finite direct products. Looking at the dual Stone spaces of Boolean algebras of subsets of a given topological algebra, we find a simple characterization of when the dual space admits a natural structure of topological algebra; the characterization is given in terms of how the Boolean algebra behaves under the inverses of the operation evaluation mappings of each arity. This provides an alternative, which is presented in the form of an equivalence of categories, to the duality theory of M. Gehrke. As an application, a Stone quotient of a Stone topological algebra that is residually in a given Stone pseudovariety is shown to be also residually in it, thereby extending the corresponding result of M. Gehrke for the Stone pseudovariety of all finite algebras over discrete signatures, which was recently extended to topological signatures jointly by the authors and H. Goulet-Ouellet. The residual closure of a Stone pseudovariety is thus a Stone pseudovariety, and these are precisely the Stone analogues of varieties. A Birkhoff type theorem for Stone varieties is also established and it is shown how Reiterman’s theorem can be derived from it.

On the interaction of the Coxeter transformation and the rowmotion bijection

Let P be a finite poset and L the associated distributive lattice of order ideals of P . Let \rho denote the rowmotion bijection of the order ideals of P viewed as a permutation matrix and C the Coxeter matrix for the incidence algebra kL of L . Then, we show the identity (\rho^{-1}C)^{2}=\mathrm{id} , as was originally conjectured by Sam Hopkins. Recently, it was noted that the rowmotion bijection is a special case of the much more general grade bijection R that exists for any Auslander regular algebra. This motivates to study the interaction of the grade bijection and the Coxeter matrix for general Auslander regular algebras. For the class of higher Auslander algebras coming from n -representation finite algebras, we show that (R^{-1}C)^{2}=\mathrm{id} if n is even and (R^{-1}C+\mathrm{id})^{2}=0 when n is odd.

Young walls and graded dimension formulas for finite quiver Hecke algebras of types A2n−1(2), Dn(1) and Bn(1)

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-[Formula: see text] Fock space representations of quantum affine algebras of types [Formula: see text], [Formula: see text] and [Formula: see text].

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

Finite algebras with Hom-sets of polynomial size

We provide an internal characterization of those finite algebras (i.e., algebraic structures) A \mathbf {A} such that the number of homomorphisms from any finite algebra X \mathbf {X} to A \mathbf {A} is bounded from above by a polynomial in the size of X \mathbf {X} . Namely, an algebra A \mathbf {A} has this property if, and only if, no subalgebra of A \mathbf {A} has a nontrivial strongly abelian congruence. We also show that the property can be decided in polynomial time for algebras in finite signatures. Moreover, if A \mathbf {A} is such an algebra, the set of all homomorphisms from X \mathbf {X} to A \mathbf {A} can be computed in polynomial time given X \mathbf {X} as input. As an application of our results to the field of computational complexity, we characterize inherently tractable constraint satisfaction problems over fixed finite structures, i.e., those that are tractable and remain tractable after expanding the fixed structure by arbitrary relations or functions.

Shard Theory for g-Fans

Abstract For a finite dimensional algebra $A$, the notion of $g$-fan $\Sigma (A)$ is defined from two-term silting complexes of $\textsf{K}^{\textrm{b}}(\textsf{proj} A)$ in the real Grothendieck group $K_{0}(\textsf{proj} A)_{\mathbb{R}}$. In this paper, we discuss the theory of shards to $\Sigma (A)$, which was originally defined for a hyperplane arrangement. We establish a correspondence between the set of join-irreducible elements of torsion classes of $\textsf{mod}A$ and the set of shards of $\Sigma (A)$ for $g$-finite algebra $A$. Moreover, we show that the semistable region of a brick of $\textsf{mod}A$ is exactly given by a shard. We also give a poset isomorphism of shard intersections and wide subcategories of $\textsf{mod}A$.

The space of tracial states on a [formula omitted]-algebra

We give a simple and elementary proof that the tracial state space of a unital C∗-algebra is a Choquet simplex, using the center-valued trace on a finite von Neumann algebra.

The Structure of the Unit Group of $\mathbb{F}_{3^{k}}(C_{3}\times D_{2n})$ and Finite Group Algebras of Groups of Order $42$

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p>0$ with $|\mathbb{F}_{q}|=q=p^{k}$ and $\mathcal{U}(\mathbb{F}_{q}G)$ be the unit group of the group algebra $\mathbb{F}_{q}G$ for some group $G$. There are $6$ groups of order $42$ up to isomorphism. In this paper, we provide a characterization of $\mathcal{U}(\mathbb{F}_{3^{k}}(C_{3}\times D_{2n}))$ and establish the structures of the unit groups of some finite group algebras of groups of order $42$.

Coalgebraic methods for Ramsey degrees of unary algebras

Abstract In this paper, we prove the existence of small and big Ramsey degrees of classes of finite unary algebras in an arbitrary (not necessarily finite) algebraic language Ω. Our results generalize some Ramsey-type results of M. Sokić concerning finite unary algebras over finite languages. To do so, we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat unary algebras as Eilenberg-Moore coalgebras for a functor with comultiplication, and using pre-adjunctions transport the Ramsey properties, we are interested in from the category of finite or countably infinite chains of order type ω. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

Quasi-countable inverse semigroups as metric spaces, and the uniform Roe algebras of locally finite inverse semigroups

Given any quasi-countable, in particular, any countable inverse semigroup S , we introduce a way to equip S with a proper and right subinvariant extended metric. This generalizes the notion of proper, right invariant metrics for discrete countable groups. Such a metric is shown to be unique up to bijective coarse equivalence of the semigroup, and hence depends essentially only on S . This allows us to unambiguously define the uniform Roe algebra of S , which we prove can be realized as a canonical crossed product of \ell^{\infty}(S) and S . We relate these metrics to the analogous metrics on Hausdorff étale groupoids. Using this setting, we study those inverse semigroups with asymptotic dimension 0 . Generalizing results known for groups, we show that these are precisely the locally finite inverse semigroups and are further characterized by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in the group case, having a finite uniform Roe algebra is strictly weaker and is characterized by S being locally \mathcal{L} -finite, and equivalently sparse as a metric space.

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

Ontological description method for building service-oriented distributed learning systems

The subject of the research is the analysis and justification of the use of software engineering processes based on ontologies of data access and representation of access to knowledge bases and knowledge reuse in the language of algebra of finite predicates, which is a class of descriptive logics. This approach allows you to use the advantages of ontologies and logic programming in the refactoring of distributed distance learning systems. The purpose of the work is to create a formalism for describing the interaction of services and algorithms for building web service interfaces for the implementation of an effective SOA system using the paradigm of access to data based on ontologies. To achieve the goal, the following tasks were solved: an analysis of the main types of interaction structures of SOA services, an overview of implementations of systems working with a family of descriptive logic languages, namely, algebra of finite predicates, and a description of the structure of the ontology necessary for the operation of service-oriented systems, which has the markup of the Semantic Web standard using finite predicate algebra equations. The task is an urgent task of ensuring the organization of interaction of Internet resources in the development of distributed virtual learning systems, and such interaction is enabled by the service-oriented architecture of software development. The article examines modern technologies of the Semantic Web and their role in ensuring the creation of applied software systems of distributed learning resources built using distributed interactive services. The considered methods and materials include methods of algebra of finite predicates, theory of algorithms, object-oriented design, theory of unification. The results. The semantics of the SOA system is obtained, which makes it possible to describe SOA systems in the languages of descriptive logic. The effectiveness of the mathematical formalism of the algebra of finite predicates for the tasks of logical analysis of applied ontologies, the use of knowledge reuse methods, and the description of service-oriented systems is shown. The necessity of using algorithms for automatic construction of web interfaces is shown. The conclusions of the work highlight the advantages of the researched solution, namely, algorithms for the automated construction of web service interfaces for SOA architecture, which make it possible to automatically build web service interfaces. Algorithms differ in a more general model and speed of work and require an assessment of complexity.

A Method for Specifying Complete Signature Randomization and an Algebraic Algorithm Based on It

To eliminate the limitations of signature randomization in known algebraic algorithms with a hidden group, the security of which is based on the computational complexity of solving large systems of power equations, a method for ensuring complete randomization is proposed. Based on this method, a new algorithm of the indicated type was developed, using a four-dimensional finite non-commutative associative algebra as an algebraic basis. We obtained estimates of the security of algorithms to direct attacks as well as from attacks based on known signatures, which confirm the effectiveness of the proposed signature randomization method. Due to the relatively small size and signature of the public and private keys, the developed algorithm is of interest as a potential practical post-quantum digital signature scheme.

The spectrum of Grothendieck monoid: Classifying Serre subcategories and reconstruction theorem

The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. For this, we determine the Grothendieck monoid of the category of coherent sheaves on a smooth projective curve. By using (iii), we introduce a topology on the set of Serre subcategories. As a consequence, we recover the topology of a noetherian scheme from the Grothendieck monoid.