A Classification of Two-dimensional Endo-commutative Algebras over F_2

We introduce a new class of algebras called endo-commutative algebras in which the square mapping preserves multiplication, and provide a complete classification of endo-commutative algebras of dimension 2 over the field F_2 of two elements. We list all multiplication tables of the algebras up to isomorphism. This clarifies the difference between commutativity and endo-commutativity of algebras.Comment: 19 page

A classification of 2-dimensional endo-commutative straight algebras of rank 1 over a non-trivial field

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of 2-dimensional endo-commutative straight algebras of rank one over an arbitrary non-trivial field, where a straight algebra of dimension 2 satisfies the condition that there exists an element $x$ such that $x$ and $x^2$ are linearly independent. We list all multiplication tables of the algebras up to isomorphism.Comment: 10 pages. arXiv admin note: text overlap with arXiv:2304.1251

Classification of Zeropotent Algebras of Dimension 3 over mathbbRmathbb{R} (Logic, Language, Algebraic system and Related Areas in Computer Science)

This is a summary of our results [Y. Kobayashi, K. Shirayanagi, S.-E. Takahasi and M. Tsukada, Classification of three-dimensional zeropotent algebras over an algebraically closed field, Comm. Algebra, Vol. 45 (12), 5037-5052, 2017.] and [K. Shirayanagi, S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of three-dimensional zeropotent algebras over the real number field, Comm. Algebra, Vol. 46 (11), 4665-4681, 2018.]

Bender-Knuth transformation from a perspective of hives (Computer Algebra --Theory and its Applications)

Kostka numbers Kλμ are non-negative integers indexed by two partitions λ and μ.They equal to the number of semistandard tableaux of shape入andweightμ. Also they equal to the number of K-hives with boundary edge label determined by λ and μ.Their basic property is Kλμ=Kλσ(μ) for any element a of the symmetric group. This is proved by constructing a bijection called the Bender-Knuth transformation between semistandard tableaux. In this paper, we give a perspective of the Bender-Knuth transformation through the hive model

Abstract Floating point Grijbner bases

Bracket coefficients for polynomials are introduced. These are like specific precision floating point numbers together with error terms. Working in terms of bracket coefficients, an algorithm that computes a Grobner basis with floating point coefficients is presented, and a new criterion for determining whether a bracket coefficient is zero is proposed. Given a finite set F of polynomials with real coefficients, let G, be the result of the algorithm for F and a precision value p, and G be a true Grobner basis of F. Then, as p approaches infinity, G, converges to G coefficientwise. Moreover, there is a precision M such that if p 3 M, then the sets of monomials with non-zero coefficients of G, and G are exactly the same. The practical usefulness of the algorithm is suggested by experimental results. 1

A new Gröbner basis conversion method based on stabilization techniques

AbstractWe propose a new method for converting a Gröbner basis w.r.t. one term order into a Gröbner basis w.r.t. another term order by using the algorithm stabilization techniques proposed by Shirayanagi and Sweedler. First, we guess the support of the desired Gröbner basis from a floating-point Gröbner basis by exploiting the supportwise convergence property of the stabilized Buchberger’s algorithm. Next, assuming this support to be correct, we use linear algebra, namely, the method of indeterminate coefficients to determine the exact values for the coefficients. Related work includes the FGLM algorithm and its modular version. Our method is new in the sense that it can be thought of as a floating-point approach to the linear algebra method. The results of Maple computing experiments indicate that our method can be very effective in the case of non-rational coefficients, especially the ones including transcendental constants