Tropical Roots and Their Multiplicities on Subsemigroup of Order-Decreasing and Order-Preserving Full Transformation

Abstract

Introduction: Let be ordered finite set. The tropical geometry was utilized to analyze the subsemigroup of order-decreasing and order-preserving full transformation semigroups, denoted as . The elements of classical algebra within were transformed into tropical polynomials, allowing for the determination of tropical roots and their multiplicities through the tropical curve, which was visualized using GeoGebra. Aims: The primary aim of this study is to establish a connection between tropical geometry and the algebraic structure of Sub semigroup of Order-Decreasing and Order-Preserving full transformation semigroups Materials and Methods: The study begins by extracting the elements of the from detailing its structure and properties. Elements of are then converted into tropical polynomials using tropical algebraic operations, specifically tropical addition and multiplication. The study examines these tropical polynomials and computes the multiplicities of their roots using GeoGebra software to illustrate them graphically. Theorems are formulated to describe the properties of the tropicalized elements, with proofs provided to support these results. Furthermore, tropical curves are analyzed to observe the structure of the transformations, and examples are presented to illustrate and confirm the theoretical findings. Results: The findings reveal that, although the tropical roots differ for each value of , the corresponding multiplicities remain the same, and the sum of these multiplicities equals the degree of the associated classical polynomial. Moreover, it was observed that for , all multiplicities have a height of 1, whereas for , the height of the multiplicities is 2. Additionally, for , the multiplicities follow the pattern . Conclusion: This work offers insightful results and lays a solid foundation for further exploration in the area of tropical geometry and transformation semigroups, especially for those interested in the interplay between algebraic and tropical structures.