On Gröbner Bases via Alternating Semigroup

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 (𝐴𝜂 𝜍). The study specifically focuses on the exploration of Gröbner bases in the context of 𝐴𝜂 𝜍, 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 (𝐴𝜂 𝜍) is a lexicographic order, whereas the converse does not hold. Additionally, 𝑀 (𝐴𝜂 𝜍) 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.