Structure Of Finite Algebras Research Articles

On polynomial completeness properties of finite Mal’cev algebras

Polynomial completeness results aim at characterizing those functions that are induced by polynomials. Each polynomial function is congruence preserving, but the opposite need not be true. A finite algebraic structure [Formula: see text] is called strictly 1-affine complete if every unary partial function from a subset of [Formula: see text] to [Formula: see text] that preserves the congruences of [Formula: see text] can be interpolated by a polynomial function of [Formula: see text]. The problem of characterizing strictly 1-affine complete finite Mal’cev algebras is still open. In this paper, we extend the characterization by Aichinger and Idziak of strictly 1-affine complete expanded groups to finite congruence regular Mal’cev algebras.

Algebras from finite group actions and a question of Eilenberg and Schützenberger

In 1976 S. Eilenberg and M.-P. Schützenberger posed the following diabolical question: if A is a finite algebraic structure, Σ is the set of all identities true in A, and there exists a finite subset F of Σ such that F and Σ have exactly the same finite models, must there also exist a finite subset F′ of Σ such that F′ and Σ have exactly the same finite and infinite models? (That is, must the identities of A be “finitely based”?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the “inherently nonfinitely based” algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Schützenberger. As a corollary, we give the first known examples of inherently nonfinitely based “automatic algebras” constructed from group actions.

Прямые степени алгебраических систем и уравнения над ними

We study systems of equations over graphs, posets and matroids. We give the criteria when a direct power of such algebraic structures is equationally Noetherian. Moreover, we prove that any direct power of any finite algebraic structure is weakly equationally Noetherian.

C#-IDEALS OF LIE ALGEBRAS

Let $L$ be a finite dimensional Lie algebra. A subalgebra $H$ of $L$ is called a $c^{#}$-ideal of $L$, if there is an ideal $K$ of $L$ with $L=H+K$ and $Hcap K$ is a $CAP$-subalgebra of $L$. This is analogous to the concept of a $c^{#}$-normal subgroup of a finite group. Now, we consider the influence of this concept on the structure of finite dimentional Lie algebras.

On generic complexity of theories of finite algebraic structures

This article is devoted to investigation of generic complexity of universal and elementary theories of finite algebraic structures with universal set with more than one element of finite predicate signature with equality. We prove that there are no polynomial strongly generic algorithm, recognizing any such theory, provided P = BPP and P ≠ NP (P ≠ PSPACE). The author is supported by Russian Science Foundation, grant 19-11-00209.

On the algebraic structure of the Moore–Penrose inverse of a polynomial matrix

AbstractThis work establishes the connection between the finite and infinite algebraic structure of singular polynomial matrices and their Moore–Penrose (MP) inverse. The uniqueness of the MP inverse leads to the assumption that such a relation must exist. It is proved that the MP inverse of a singular polynomial matrix has no finite zeros and its finite poles are fully determined. Furthermore, the existence of a correspondence between the infinite pole/zero structure of any singular polynomial matrix and its MP inverse is proved to exist. Finally, it is shown that the minimal indices of the two aforementioned matrices are connected as well.

Hilbert functions of certain rings of invariants via representations of the symmetric groups (with an appendix by Dejan Govc)

In this paper we study rings of invariants arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form K[U]Γ where Γ is a product of general linear groups over a field K of characteristic zero, and U is a finite dimensional rational representation of Γ. We will calculate the Hilbert series of such rings using the representation theory of the symmetric groups and Schur-Weyl duality. We focus on the case where U=End(W⊕k) and Γ=GL(W) and on the case where U=End(V⊗W) and Γ=GL(V)×GL(W), though the methods introduced here can also be applied in more general framework. For the two aforementioned cases we calculate the Hilbert function of the ring of invariants in terms of Littlewood-Richardson and Kronecker coefficients. When the vector spaces are of dimension 2 we also give an explicit calculation of this Hilbert series, using Mathematica (see the appendix by Dejan Govc).

A novel discrete image encryption algorithm based on finite algebraic structures

The arithmetic properties of a finite field have a remarkable impact on the security features of symmetric and asymmetric cryptosystems. Conventionally, in modern symmetric-key cryptosystems, a 256 elements Galois field depending on a single 8 degree primitive irreducible polynomial over ℤ2 is used for designing an S-box, the main nonlinear component in AES. The experience of S-box based image encryption schemes is not appreciable than chaotic systems based methods. In this study, we improve the S-box based image encryption algorithms by the usage of all 16 distinct degree 8 primitive irreducible polynomials over ℤ2 and by introducing a new role of the ring ℤn of integers modulo n in the permutation steps. The strength of this novel 2-algebraic structures based image encryption scheme is evaluated by some statistical analyses and a significant performance level is achieved. A comparison of encryption quality with some of the recent chaos-based image encryption algorithms is given and evidently the newly launched scheme spectacles high performance. Thus this new development in image encryption methods may provide an alternative to chaos dependent image encryption.

A generic algorithm for the identity problem in finite groups and monoids

Kapovich, Myasnikov, Schupp and Shpilrain in 2003 developed generic approach to algorithmic problems, which considers an algorithmic problem on “most” of the inputs instead of the entire domain and ignores it on the rest of inputs. This approach can be applied to algorithmic problems, which are hard in the classical sense. The problem of checking identities in algebraic structures is the one of the most fundamental problem in algebra. For finite algebraic structures this problem can be decidable in polynomial time, or hard (co-NP-complete). In this paper we present a generic polynomial algorithm for the identity problem in all finite groups and monoids with elements of period greater than 1. The work is supported by Mathematical Center in Akademgorodok, the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613.

Deciding Quantifier-free Definability in Finite Algebraic Structures

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present a decision algorithm based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.

Eccentric connectivity index of identity graph of cyclic group and finite commutative ring with unity

Research on graph associated with a finite algebraic structure has attracted many attentions. On the other hand, eccentric connectivity index is an interesting topic and many studies have been reported. For simple connected graph G, let e(v) denoted the eccentricity of vertex v and deg(v) denoted the degree of vertex v in G. Eccentric connectivity index of G is defined as the sum of all e(v)deg(v), for any v in G. We focus the study on determining eccentricity connectivity index of identity graph of cyclic group and finite commutative ring with unity. We present the exact formula for eccentricity connectivity index of identity graph of these two algebraic structures.

Combinatorics of X-variables in finite type cluster algebras

We compute the number of X-variables (also called coefficients) of a cluster algebra of finite type when the underlying semifield is the universal semifield. For classical types, these numbers arise from a bijection between coefficients and quadrilaterals (with a choice of diagonal) appearing in triangulations of certain marked surfaces. We conjecture that similar results hold for cluster algebras from arbitrary marked surfaces, and obtain corollaries regarding the structure of finite type cluster algebras of geometric type.

Fully Homomorphic Cipher Based on Finite Algebraic Structures

A way to increase the robustness of a cryptographic algorithm toward unauthorized inversion can be obtained through application of non-commutative or non-associative algebraic structures. In this regard, data security became a great issue in adaptation of cloud computing over Internet. While in the traditional encryption methods, security to data in storage state and transmission state is provided, in cloud data processing state, decryption of data is assumed, data being available to cloud provider. In this paper, we propose a special homomorphism between self-distributed and non-associative algebraic structures, which can stand as a premise to construct a homomorphic encryption algorithm aimed at the cloud data security in processing state. Homomorphic encryption so developed will allow users to operate encrypted data directly bypassing the decryption.

Certain numerical results in non-associative structures

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer nge 2, the nth-commutativity degree of a finite algebraic structure S, denoted by P_n(S), is the probability that for chosen randomly two elements x and y of S, the relator x^ny=yx^n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.

Finite dimensional estimation algebras with state dimension 3 and rank 2, Mitter conjecture

ABSTRACT In this paper, we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and under-determined partial differential equations. It is proved that if the estimation algebra contains a degree two polynomial, then the Wong Ω-matrix must be a constant matrix. Moreover, all functions in the estimation algebra must be linear functions.

Selfinjective algebras without short cycles of indecomposable modules

We describe the structure of finite dimensional selfinjective algebras over an arbitrary field without short cycles of indecomposable modules.

Relativised Homomorphism Preservation at the Finite Level

In this article, we investigate the status of the homomorphism preservation property amongst restricted classes of finite relational structures and algebraic structures. We show that there are many homomorphism-closed classes of finite lattices that are definable by a first-order sentence but not by existential positive sentences, demonstrating the failure of the homomorphism preservation property for lattices at the finite level. In contrast to the negative results for algebras, we establish a finite-level relativised homomorphism preservation theorem in the relational case. More specifically, we give a complete finite-level characterisation of first-order definable finitely generated anti-varieties relative to classes of relational structures definable by sentences of some general forms. When relativisation is dropped, this gives a fresh proof of Atserias’s characterisation of first-order definable constraint satisfaction problems over a fixed template, a well known special case of Rossman’s Finite Homomorphism Preservation Theorem.

Finite Dimensional Estimation Algebras with State Dimension 3 and rank 2, I: Linear Structure of Wong Matrix

In this paper we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the Euler operator and underdetermined partial differential equations. The structure of the Wong $\Omega$-matrix is shown to be linear. The fundamental strategy we use in this paper to prove these results is to show that if they were not true, then infinite sequences could be constructed in the finite dimensional estimation algebra.

Descent, fields of invariants, and generic forms via symmetric monoidal categories

Let W be a finite dimensional algebraic structure (e.g. an algebra) over a field K of characteristic zero. We study forms of W by using Deligne's Theory of symmetric monoidal categories. We construct a category CW, which gives rise to a subfield K0⊆K, which we call the field of invariants of W. This field will be contained in any subfield of K over which W has a form. The category CW is a K0-form of RepK¯(Aut(W)), and we use it to construct a generic form W˜ over a commutative K0-algebra BW (so that forms of W are exactly the specializations of W˜). This generalizes some generic constructions for central simple algebras and for H-comodule algebras. We give some concrete examples arising from associative algebras and H-comodule algebras. As an application, we also explain how one can use the construction to classify two-cocycles on some finite dimensional Hopf algebras.

On selfinjective algebras of finite representation type with maximal almost split sequences

We investigate the structure of finite dimensional selfinjective algebras of finite representation type over an arbitrary field having almost split sequences of modules whose middle term admits the maximal possible number of indecomposable direct summands.