On Gröbner Bases via Alternating Semigroup
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Date
2024-12-23
Journal Title
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Publisher
Federal University of Oye-Ekiti , Nigeria.
Abstract
Let Ω𝜂 = {1,2 , … , 𝜂} be a finite set ordered in standard way. This paper presents an
in-depth investigation into the application of Gröbner bases within the framework of
alternating semigroups (𝐴𝜂𝜍). The study specifically focuses on the exploration of Gröbner bases in the context of 𝐴𝜍𝜂, which forms a subset of the symmetric inverse semigroup (𝐶𝜂 ). Key results derived concerning monomial orderings, including Lexicographic (LEX), Graded Lexicographic (GRLEX), and Graded Reverse Lexicographic (GRVLEX) orders. Monomials are degenerated, and computational techniques employed to achieve the outcomes. The broader significance lies in the potential applications of these results in areas such as combinatorial optimization, symbolic computation, and automated theorem proving. By demonstrating the practical utility of monomial degenerations and computational techniques, this research opens
new avenues for efficient problem-solving in both theoretical and applied mathematical domains. A notable observation is that every graded lexicographic ordering of 𝑀(𝐴𝜍𝜂) is a lexicographic order, whereas the converse does not hold. Additionally, 𝑀(𝐴𝜍𝜂) adheres to graded reverse lexicographic order if and only if the index of the first entry of the first monomial is smaller than that of the second monomial. Some classical examples with their graphical representations included to substantiate the theoretical results.
Description
Bakare, G. N., Ajibaye, O. K., Usamot. I. F., Ibrahim, G. R., Adeshola A. D. (2024): On Gröbner Bases via Alternating Semigroup. Journal of Pure and Applied Sciences (FUOYE) 9(3) ISSN: 2616 – 1419; 83-92. Published by Federal University of Oye-Ekiti , Nigeria.
Keywords
Monomial, Monomial ordering, Lexicographic, Graded Lexicographic, Graded reverse Lexicographic
Citation
Bakare, G. N., Ajibaye, O. K., Usamot. I. F., Ibrahim, G. R., Adeshola A. D. (2024): On Gröbner Bases via Alternating Semigroup. Journal of Pure and Applied Sciences (FUOYE) 9(3) ISSN: 2616 – 1419; 83-92. Published by Federal University of Oye-Ekiti , Nigeria.