An Analytical Approach to Bimonotone Linear Inequalities and Sublattice Structures

Abstract

This paper presents a comprehensive analytical approach to the study of bimonotone linear inequalities and their relationship with sublattice structures in Rⁿ. Bimonotone inequalities, which are linear constraints characterized by coordinate-wise monotonicity, naturally arise in optimization problems, economics, and combinatorial geometry. We explore the geometric properties of convex sublattices and demonstrate how they can be efficiently represented as the solution sets of systems of bimonotone inequalities. By analyzing the algebraic and geometric structures of these sublattices, we provide new insights into their behavior under various operations, such as intersection and projection. Additionally, the paper discusses the application of these concepts in optimization, game theory, and machine learning, where monotonicity constraints are commonly employed. The results contribute to a deeper understanding of how bimonotone inequalities define convex sets with lattice structures, offering a valuable tool for solving high-dimensional optimization problems with monotonicity constraints