Fully Invariant And Verbal Congruence Relations, 2017 Iowa State University

#### Fully Invariant And Verbal Congruence Relations, Clifford Bergman, Joel Berman

*Clifford Bergman*

A congruence relation *θ* on an algebra A is fully invariant if every endomorphism of A preserves *θ*. A congruence *θ* is verbal if there exists a variety V" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">VV such that *θ* is the least congruence of Asuch that A/θ∈V" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction ...

Automorphism-Primal Algebras Generate Verbose Varieties, 2017 Iowa State University

#### Automorphism-Primal Algebras Generate Verbose Varieties, Clifford Bergman

*Clifford Bergman*

A finite algebra is called automorphism-primal if its clone of term operations coincides with all operations that preserve its automorphisms. We prove that the variety generated by an automorphism-primal algebra is verbose, that is, on every member algebra, every fully invariant congruence is verbal. The proof is a nice application of the theory of natural dualities as developed by Davey et al.

Commutative Idempotent Groupoids And The Constraint Satisfaction Problem, 2017 Iowa State University

#### Commutative Idempotent Groupoids And The Constraint Satisfaction Problem, Clifford Bergman, David Failing

*Clifford Bergman*

A restatement of the Algebraic Dichotomy Conjecture, due to Maroti and McKenzie, postulates that if a finite algebra A possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI groupoid) will be tractable. It is known that every semilattice (i.e., an associative CI groupoid) is tractable. A groupoid identity is of Bol–Moufang type if the same three variables appear on either side, one of the variables ...

Experimenting With The Identity (Xy)Z = Y(Zx), 2017 Iowa State University

#### Experimenting With The Identity (Xy)Z = Y(Zx), Irvin Roy Hentzel, David P. Jacobs, Sekhar V. Muddana

*Irvin Roy Hentzel*

An experiment with the nonassociative algebra program Albert led to the discovery of the following surprising theorem. *Let G be a groupoid satisfying the identity (xy)z = y(zx). Then for products in G involving at least five elements, all factors commute and associate.* A corollary is that any semiprime ring satisfying this identity must be commutative and associative, generalizing a known result of Chen.

Semiprime Locally(-1, 1) Ring With Minimal Condition, 2017 Iowa State University

#### Semiprime Locally(-1, 1) Ring With Minimal Condition, Irvin R. Hentzel, H. F. Smith

*Irvin Roy Hentzel*

Let L be a left ideal of a right alternative ring A with characteristic ::/=2. If L is maximal and nil, then L is a two-sided ideal. If L is minimal, then it is either a two-sided ideal, or the ideal it generates is contained in the right nucleus of A. In particular, if A is prime, then a minimal left ideal of A must be a two-sided ideal. Let A be a semiprime locally (-1, 1) ring with characteristic ::1=2, 3. Then A is isomorphic to a subdirect sum of an alternative ring, a strong (-1, 1) ring ...

Minimal Identities Of Bernstein Alegebras, 2017 Iowa State University

#### Minimal Identities Of Bernstein Alegebras, Irvin R. Hentzel, Ivan Correa, Luiz Antionio Peresi

*Irvin Roy Hentzel*

We construct the minimal identities for Bernstein algebras, exceptional Bernstein algebras and normal Bernstein algebras. We use the technique of processing identities via the representation of the symmetric groups. The computer algorithms for creating the standard tableaus and the integral representations are summarized.

Counterexamples In Nonassociative Algebra, 2017 Iowa State University

#### Counterexamples In Nonassociative Algebra, Irvin R. Hentzel, Luiz Antonio Peresi

*Irvin Roy Hentzel*

We present a method of constructing counterexamples in nonassociative algebra. The heart of the computation is constructing a matrix of identities and reducing this matrix (usually very sparse) to row canonical form. The example is constructed from the entries in one column of this row canonical form. While this procedure is not polynomial in the degree of the identity, several shortcuts are listed which shorten calculations. Several examples are given.

On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, 2017 University of Wyoming

#### On Preserving Structured Matrices Using Double Bracket Operators: Tridiagonal And Toeplitz Matrices, Kenneth Driessel, Irvin R. Hentzel, Wasin So

*Irvin Roy Hentzel*

In the algebra of square matrices over the complex numbers, denotes Two problems are solved: (1) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is tridiagonal, then so is (2) Find all Hermitian matrices M which have the following property: For every Hermitian matrix A, if A is Toeplitz, then so is

On Prime Right Alternative Algebras And Alternators, 2017 Italian National Research Council

#### On Prime Right Alternative Algebras And Alternators, Giulia Maria Piacentini Cattaneo, Irvin R. Hentzel

*Irvin Roy Hentzel*

We study subvarieties of the variety of right alternative algebras over a field of characteristic t2,t3 such that the defining identities of the variety force the span of the alternators to be an ideal and do not force an algebra with identity element to be alternative. We call a member of such a variety a right alternative alternator ideal algebra. We characterize the algebras of this subvariety by finding an identity which holds if and only if an algebra belongs to the subvariety. We use this identity to prove that if R is a prime, right alternative alternator ideal ...

An Introduction To Lie Algebra, 2017 California State University – San Bernardino

#### An Introduction To Lie Algebra, Amanda Renee Talley

*Electronic Theses, Projects, and Dissertations*

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency ...

Investigation Of Finite Groups Through Progenitors, 2017 California State University, San Bernardino

#### Investigation Of Finite Groups Through Progenitors, Charles Baccari

*Electronic Theses, Projects, and Dissertations*

The goal of this presentation is to find original symmetric presentations of finite groups. It is frequently the case, that progenitors factored by appropriate relations produce simple and even sporadic groups as homomorphic images. We have discovered two of the twenty-six sporadic simple groups namely, M_{12}, J_{1 }and the Lie type group Suz(8). In addition several linear and classical groups will also be presented. We will present several progenitors including: 2^{*12}: 2^{2} x (3 : 2), 2^{*11}: PSL_{2}(11), 2^{*5}: (5 : 4) which have produced the homomorphic images: M_{12} : 2, Suz(8 ...

Groups Of Matrices That Act Monopotently, 2017 Colby College

#### Groups Of Matrices That Act Monopotently, Joshua D. Hews, Leo Livshits

*Electronic Journal of Linear Algebra*

In the present article, the authors continue the line of inquiry started by Cigler and Jerman, who studied the separation of eigenvalues of a matrix under an action of a matrix group. The authors consider groups \Fam{G} of matrices of the form $\left[\small{\begin{smallmatrix} G & 0\\ 0& z \end{smallmatrix}}\right]$, where $z$ is a complex number, and the matrices $G$ form an irreducible subgroup of $\GL(\C)$. When \Fam{G} is not essentially finite, the authors prove that for each invertible $A$ the set $\Fam{G}A$ contains a matrix with more than one eigenvalue. The ...

On One-Sided (B;C)-Inverses Of Arbitrary Matrices, 2017 Universidad Politécnica de Valencia

#### On One-Sided (B;C)-Inverses Of Arbitrary Matrices, Julio Benitez, Enrico Boasso, Hongwei Jin

*Electronic Journal of Linear Algebra*

In this article, one-sided $(b, c)$-inverses of arbitrary matrices as well as one-sided inverses along a (not necessarily square) matrix, will be studied. In addition, the $(b, c)$-inverse and the inverse along an element will be also researched in the context of rectangular matrices.

Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, 2017 Universidad de La Serena

#### Nilpotent Linear Transformations And The Solvability Of Power-Associative Nilalgebras, Ivan Correa, Irvin R. Hentzel, Pedro Pablo Julca, Luiz Antonio Peresi

*Irvin Roy Hentzel*

We prove some results about nilpotent linear transformations. As an application we solve some cases of Albert’s problem on the solvability of nilalgebras. More precisely, we prove the following results: commutative power-associative nilalgebras of dimension n and nilindex n − 1 or n − 2 are solvable; commutative power-associative nilalgebras of dimension 7 are solvable.

Nuclear Elements Of Degree 6 In The Free Alternative Algebra, 2017 Iowa State University

#### Nuclear Elements Of Degree 6 In The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi

*Irvin Roy Hentzel*

We construct five new elements of degree 6 in the nucleus of the free alternative algebra. We use the representation theory of the symmetric group to locate the elements. We use the computer algebra system ALBERT and an extension of ALBERT to express the elements in compact form and to show that these new elements are not a consequence of the known degree-5 elements in the nucleus. We prove that these five new elements and four known elements form a basis for the subspace of nuclear elements of degree 6. Our calculations are done using modular arithmetic to save memory ...

Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), 2017 Universidad de Chile

#### Solvability Of Commutative Right-Nilalgebras Satisfying (B (Aa)) A= B ((Aa) A), Ivan Correa, Alicia Labra, Irvin R. Hentzel

*Irvin Roy Hentzel*

We study commutative right-nilalgebras of right-nilindex four satisfying the identity (b(aa))a = b((aa)a). Our main result is that these algebras are solvable and not necessarily nilpotent. Our results require characteristic ≠ 2, 3, 5.

Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, 2017 University of Saskatchewan

#### Invariant Nonassociative Algebra Structures On Irreducible Representations Of Simple Lie Algebras, Murray Bremner, Irvin R. Hentzel

*Irvin Roy Hentzel*

An irreducible representation of a simple Lie algebra can be a direct summand of its own tensor square. In this case, the representation admits a nonassociative algebra structure which is invariant in the sense that the Lie algebra acts as derivations. We study this situation for the Lie algebra sl(2).

The Nucleus Of The Free Alternative Algebra, 2017 Iowa State University

#### The Nucleus Of The Free Alternative Algebra, Irvin R. Hentzel, L. A. Peresi

*Irvin Roy Hentzel*

We use a computer procedure to determine a basis of the elements of degree 5 in the nucleus of the free alternative algebra. In order to save computer memory, we do our calculations over the field Z103. All calculations are made with multilinear identities. Our procedure is also valid for other characteristics and for determining nuclear elements of higher degree.

Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, 2017 Iowa State University

#### Rings With (A, B, C) = (A, C, B) And (A, [B, C]D) = 0: A Case Study Using Albert, Irvin R. Hentzel, D. P. Jacobs, Erwin Kleinfeld

*Irvin Roy Hentzel*

Albert is an interactive computer system for building nonassociative algebras [2]. In this paper, we suggest certain techniques for using Albert that allow one to posit and test hypotheses effectively. This process provides a fast way to achieve new results, and interacts nicely with traditional methods. We demonstrate the methodology by proving that any semiprime ring, having characteristic ≠ 2, 3, and satisfying the identities (a, b, c) - (a, c, b) = (a, [b, c], d) = 0, is associative. This generalizes a recent result by Y. Paul [7].

A Variety Containing Jordan And Pseudo-Composition Algebras, 2017 Iowa State University

#### A Variety Containing Jordan And Pseudo-Composition Algebras, Irvin R. Hentzel, Luiz Antonio Peresi

*Irvin Roy Hentzel*

We consider 3-Jordan algebras, i.e., the nonassociative commutative algebras satisfying (x^3 y)x=x^3(yx). The variety of 3-Jordan algebras contains all Jordan algebras and all pseudo-composition algebras. We prove that a simple 3-Jordan algebra with idempotent is either a Jordan algebra or a pseudo-composition algebra.