Class Of Algebras Research Articles

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.

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

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.

Semilinear clannish algebras

AbstractWe define a class of associative algebras generalizing ‘clannish algebras’, as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well‐known ‘string algebras’ introduced by Butler and Ringel. Our main result is the classification of finite‐dimensional indecomposable modules for these new algebras.

Parabolic recursions for Kazhdan–Lusztig polynomials and the hypercube decomposition

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n$$\\end{document}, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of q-derived Kazhdan–Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski–Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants of the combinatorial invariance conjecture and verify their validity for permutation groups.

Skew power series rings over a prime base ring

In this paper, we investigate the structure of skew power series rings of the form S=R[[x;σ,δ]], where R is a complete, positively filtered ring and (σ,δ) is a skew derivation respecting the filtration. Our main focus is on the case in which σδ=δσ, and we aim to use techniques in non-commutative valuation theory to address the long-standing open question: if P is an invariant prime ideal of R, is PS a prime ideal of S? When R has characteristic p, our results reduce this to a finite-index problem. We also give preliminary results in the “Iwasawa algebra” case δ=σ−idR in arbitrary characteristic. A key step in our argument will be to show that for a large class of Noetherian algebras, the nilradical is “almost” (σ,δ)-invariant in a certain sense.

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity

Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA).

New Classes of Filters in DeltaAlgebras

The main objective of this topic is to define new types of filters in algebra, such as filter and complete filter. Some theorems that establish the link between these filters were also presented and validated. Any filter in algebra is shown it is a complete filter. Furthermore, regular algebra is demonstrated as a new class of algebra. Additionally, this study introduces two new concepts such as complete ideal and normal. Also, any normal in regular algebra is demonstrated that it is a subalgebra. In addition, the connections between the complete ideal and the subalgebra is investigated. However, some instances are provided to clarify and bolster our results and findings.

Model-theoretic properties of nilpotent groups and Lie algebras

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fraïssé limit of 2-nilpotent groups of exponent p studied by Baudisch is 2-dependent and NSOP1. We prove that the class of c-nilpotent Lie algebras over an arbitrary field, in a language with predicates for a Lazard series, is closed under free amalgamation. We show that for 2<c, the generic c-nilpotent Lie algebra over Fp is strictly NSOP4 and c-dependent. Via the Lazard correspondence, we obtain the same result for c-nilpotent groups of exponent p, for an odd prime p>c.

Early Algebra Affects Peer Composition

Although research indicates that individual students benefit when they enroll in early algebra classes, evaluations of broad-based algebra acceleration often report negative effects. Using a regression discontinuity design, we replicate the positive effects of eighth-grade algebra placement on student achievement found in prior studies. We then demonstrate that eighth-grade algebra placement positively affects the achievement level of students’ classmates, as well as the experience and value added of students’ math teachers. Peer composition effects may help to reconcile the apparent contradiction between the positive effects of algebra on individual students and negative effects of algebra acceleration since these estimated effects are large enough to plausibly explain the majority of the effects of eighth-grade algebra on student test scores.

Automorphism groups of axial algebras

Axial algebras are a class of commutative non-associative algebras which have a natural group of automorphisms, called the Miyamoto group. The motivating example is the Griess algebra which has the Monster sporadic simple group as its Miyamoto group. Previously, using an expansion algorithm, about 200 examples of axial algebras in the same class as the Griess algebra have been constructed in dimensions up to about 300. In this list, we see many reoccurring dimensions which suggests that there may be some unexpected isomorphisms. Such isomorphisms can be found when the full automorphism groups of the algebras are known. Hence, in this paper, we develop methods for computing the full automorphism groups of axial algebras and apply them to a number of examples of dimensions up to 151.

Quantum geometrodynamics revived I. Classical constraint algebra

Abstract In this series of papers, we present a set of methods to revive quantum geometrodynamics which encountered numerous mathematical and conceptual challenges in its original form promoted by Wheeler and De Witt. In this paper, we introduce the regularization scheme on which we base the subsequent quantization and continuum limit of the theory. Specifically, we employ the set of piecewise constant fields as the phase space of classical geometrodynamics, resulting in a theory with finitely many degrees of freedom of the spatial metric field. As this representation effectively corresponds to a lattice theory, we can utilize well–known techniques to depict the constraints and their algebra on the lattice. We are able to compute the lattice corrections to the constraint algebra. This model can now be quantized using the usual methods of finite–dimensional quantum mechanics, as we demonstrate in the following paper. The application of the continuum limit is the subject of a future publication.&amp;#xD;

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.

Anti-C*-algebras

We introduce a class of Banach algebras that we call anti-C*-algebras. We show that the normed standard embedding of a C*-ternary ring is the direct sum of a C*-algebra and an anti-C*-algebra. We prove that C*-ternary rings and anti-C*-algebras are semisimple. We give two new characterizations of C*-ternary rings which are isomorphic to a TRO (ternary ring of operators), providing answers to a query raised by Zettl (Adv Math 48(2): 117–143, 1983), and we propose some problems for further study.

AN OUTSPREAD ON VALUED LOGIC SUPERHYPERALGEBRAS

In all classical logical algebras, only two elements can be equated to one element, and this is not possible when we want to equate more than two elements to one or more than one element. In this study, we are looking for a new idea to cover this defect. This paper considers the logic algebra structures and generalizes them to superhyper logic algebra. Indeed, we extended the axioms of logic algebra to neutrosophic superhyper logic algebras.

Axial view on pseudo-composition algebras and train algebras of rank 3

We show that pseudo-composition algebras and train algebras of rank 3 generated by idempotents are characterized as axial algebras with fusion laws derived from the Peirce decompositions of idempotents in these classes of algebras. The corresponding axial algebras are called [Formula: see text]-axial algebras, where [Formula: see text] is an element of the ground field. As a first step toward their classification, we describe the [Formula: see text]- and [Formula: see text]-generated subalgebras of such algebras.

Double wreath quasi-Hopf algebras

For a quasi-bialgebra H, we show that the category C of H-bimodules is duoidal and that the so called u-coaugmented bimonoids in C are exactly the quasi-bialgebras with a coalgebra projection. When H is a quasi-Hopf algebra with bijective antipode, we prove that the u-coaugmented bimonoids in C can also be described as what we will call double wreath quasi-Hopf algebras, objects determined by H and pre-bialgebras R within the category of Yetter-Drinfeld modules over H. A particular class of double wreath quasi-Hopf algebras is obtained by deforming with a 2-cocycle the multiplication of a Radford biproduct quasi-Hopf algebra. Other classes of this type are given by the symplectic fermion quasi-Hopf algebras.

Hyperpolyadic Structures

We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative.

Non-piecewise hereditary Nakayama algebras

Happel and Seidel gave a classification of piecewise hereditary Nakayama algebras, where the relations are given by some power of the radical.Here we explore what happens for general relations. We develop techniques for showing that a given algebra is not piecewise hereditary, illustrating them on numerous mid-sized examples. Then we observe cases where the property of being non-piecewise hereditary can be extended to other (larger) Nakayama algebras. While a complete classification remains elusive, we are able to identify two types of patterns of relations preventing piecewise heredity, indicating that for large quivers many Nakayama algebras are non-piecewise hereditary.