Theorem Of Algebra Research Articles

On an elementary proof of the fundamental theorem of algebra

On an elementary proof of the fundamental theorem of algebra

A Recurrent Proof of the Fundamental Theorem of Algebra

A Recurrent Proof of the Fundamental Theorem of Algebra

To the Bicommutant Theorem for Algebras Generated by the Symmetries of Finite Point Sets in $${\mathbb{R}}^{3}$$

To the Bicommutant Theorem for Algebras Generated by the Symmetries of Finite Point Sets in $${\mathbb{R}}^{3}$$

Ore localisation for differential graded rings; towards Goldie's theorem for differential graded algebras

We study Ore localisation of differential graded algebras. Further we define dg-prime rings, dg-semiprime rings, and study the dg-nil radical of dg-rings. Then, we define dg-essential submodules, dg-uniform dimension, and apply all this to a dg-version of Goldie's theorem on prime dg-rings.

Filtered colimit elimination from Birkhoff's variety theorem

Birkhoff's variety theorem, a fundamental theorem of universal algebra, asserts that a subclass of a given algebra is definable by equations if and only if it satisfies specific closure properties. In a generalized version of this theorem, closure under filtered colimits is required. However, in some special cases, such as finite-sorted equational theories and ordered algebraic theories, the theorem holds without assuming closure under filtered colimits. We call this phenomenon “filtered colimit elimination,” and study a sufficient condition for it. We show that if a locally finitely presentable category A satisfies a noetherian-like condition, then filtered colimit elimination holds in the generalized Birkhoff's theorem for algebras relative to A.

Full runner removal theorem for Ariki-Koike algebras

We consider the representation theory of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group Cr≀Sn. We define the addition of a runner full of beads for the abacus display of a multipartition and investigate some combinatorial properties of this operation. We focus our attention on the v-decomposition numbers, i.e. the polynomials arising from the Fock space representation of the quantum group Uv(slˆe). Using Fayers' LLT-type algorithm for Ariki-Koike algebras, we relate v-decomposition numbers for different values of e for the class of e-multiregular multipartitions, by adding a full runner of beads to each component of the abacus displays for the labelling multipartitions.

An algebraic approach to the Fundamental Theorem of Algebra

AbstractIn this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real-closed fields and quadratic forms. We show, by means of Galois theory, that $$\mathcal {F}(\sqrt{-1})$$ F ( - 1 ) is algebraically closed if $$\mathcal {F}$$ F is real-closed. Lastly, we explain the algebraic closure of $$\mathbb {R}(\sqrt{-1})=\mathbb {C}$$ R ( - 1 ) = C by demonstrating the real-closeness of $$\mathbb {R}$$ R .

Adapting the Proof of Lagrange’s Theorem into a Sequence of Group-Work Tasks

This article presents an inquiry-oriented lesson for teaching Lagrange’s theorem in abstract algebra. This lesson was developed and refined as part of a larger grant project focused on how to “Orchestrate Discussions Around Proof” (ODAP, the name of the project). The lesson components were developed and refined with attention to how well they supported active and broad student participation. By guiding student exploration of a few key example groups and structuring the exploration to identify recurrent structure, students formulate key lemmas that they can combine to prove Lagrange’s theorem.

The Arens-Calderon theorem for commutative topological algebras

The Arens-Calderon theorem for commutative topological algebras

A general mirror equivalence theorem for coset vertex operator algebras

A general mirror equivalence theorem for coset vertex operator algebras

Simulation limitations of affine cellular automata

Cellular automata are a famous model of computation, yet it is still a challenging task to assess the computational capacity of a given automaton; especially when it comes to showing negative results. In this paper, we focus on studying this problem via the notion of CA intrinsic simulation. We say that automaton A is simulated by B if each space-time diagram of A can be, after suitable transformations, reproduced by B.We study affine automata – i.e., automata whose local rules are affine mappings of vector spaces. This broad class contains the well-studied cases of linear automata. The main result of this paper shows that (almost) every automaton affine over a finite field Fp can only simulate affine automata over Fp. We discuss how this general result implies, and widely surpasses, limitations of linear and additive automata previously proved in the literature.We provide a formalization of the simulation notions into algebraic language and discuss how this opens a new path to showing negative results about the computational power of cellular automata using deeper algebraic theorems.

A Hua-type theorem for Cayley algebras

A Hua-type theorem for Cayley algebras

ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS

AbstractThe one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts $\mathrm {S5}$ and $\mathrm {MIPC}$ of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.

The reduction theorem for algebras of one-sided subshifts over arbitrary alphabets

The reduction theorem for algebras of one-sided subshifts over arbitrary alphabets

A set-theoretic approach to algebraic L-domains

AbstractIn this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.

On the schematicness of some Ore polynomials of higher order generated by homogenous quadratic relations

In this paper, we investigate the property of schematicness introduced by Van Oystaeyen and Willaert [F. Van Oystaeyen and L. Willaert, Grothendieck topology, coherent sheaves and Serre’s theorem for schematic algebras, J. Pure Appl. Algebra 104(1–3) (1995) 109–122] in the setting of skew Ore polynomials of higher order generated by homogenous quadratic relations defined by Golovashkin and Maksimov [A. V. Golovashkin and V. M. Maksimov, Skew Ore polynomials of higher orders generated by homogeneous quadratic relations, Russian Math. Surveys 53(2) (1998) 384–386]. We establish sufficient conditions to guarantee whether some of these algebras are schematic or not.

On Schur-type theorem for Leibniz 3-algebras

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group G/ζ(G) of a group G is finite, then its derived subgroup [G,G] is also finite. This theorem was proved by B. Neumann in 1951. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, n-groups, associative algebras, Lie algebras, Lie n-algebras. In 2016, L.A. Kurdachenko, J. Otal and O.O. Pypka proved an analogue of Schur theorem for Leibniz algebras: if central factor-algebra L/ζ(L) of Leibniz algebra L has finite dimension, then its derived ideal [L,L] is also finite-dimensional. Moreover, they also proved a slightly modified analogue of Schur theorem: if the codimensions of the left ζ^l (L) and right ζ^r (L) centers of Leibniz algebra L are finite, then its derived ideal [L,L] is also finite-dimensional. One of the generalizations of Leibniz algebras is the so-called Leibniz n-algebras. It is worth noting that Leibniz n-algebra theory is currently much less developed than Leibniz algebra theory. One of the directions of development of the general theory of Leibniz n-algebras is the search for analogies with other types of algebras. Therefore, the question of proving analogs of the above results for this type of algebras naturally arises. In this article, we prove the analogues of the two mentioned theorems for Leibniz n-algebras for the case n=3. The obtained results indicate the prospects of further research in this direction.

Classification of Real Division Algebras

This article aims to offer a unifying approach to the basic theory of division algebras by presenting the research of the German-American mathematician Max August Zorn, who classified alternative division algebras. In section 1 the basic theory of real division algebras is developed. Section 2 presents the Cayley-Dickson Process, which consists of constructing an extension algebra from an algebra provided with a conjugation, similar to the construction of complex numbers from real numbers. In Section 3 presents the classical division algebras R (real), C (complex), H (quaternions) and O (octonions) and mentions some of their applications. In section 4 the main theorem is presented, which establishes that the only (except isomorphism) alternative division algebras are: R, C, H and O (Zorn’s theorem). The classification theorems of associative division algebras (Frobenius) and normed division algebras (Hurwitz) are obtained as corollaries of Zorn’s theorem. Finally in section 5 applications of division algebras to Geometry, Number Theory, Classical Physics, Modern Physics, Quantum Mechanics and Cryptography are mentioned.

Schur- and Baer-type theorems for Lie and Leibniz algebras

Schur- and Baer-type theorems for Lie and Leibniz algebras

Applications of the Spectral Theorem: Utilizing Eigenvalues and Eigenvectors

This paper explores the applications and principles of both the real and complex spectral theorems, cornerstones of linear algebra and functional analysis. We begin with an overview of the spectral theorem and delve into the pivotal roles of eigenvalues and eigenvectors, culminating in a detailed proof of the theorem. A discerning comparison is made between the real and complex versions; notably, the latter seamlessly integrates with the fundamental theorem of algebra, an attribute absent in the former. The ensuing sections illuminate practical applications. The real spectral theorem emerges as instrumental in multivariable calculus, notably in the second derivative test, in data-driven techniques such as Principal Component Analysis (PCA), and in electrical network analyses via Kirchhoff’s matrix. The complex variant takes center stage in quantum mechanics, illuminating the Schrödinger Equation. This paper underscores the spectral theorem’s profound relevance in diverse theoretical and practical arenas.