A Generalized Structural Inequality: The “<<=” and “>>=” Symbols for Class-Preserving Comparison

This study introduces a new class of inequalities, denoted by the symbols “<<=” and “>>=”, which represent generalized, class-preserving comparisons between mathematical objects. Unlike traditional inequalities (“≤”, “≥”), the proposed symbols encode not only the magnitude relation but also the preservation of an intrinsic structural property (“smallness” or “bigness”) on both sides. These relations are formally defined and their theoretical properties are analyzed. Examples from various domains (numbers, functions, sets, physical models) are provided, and potential implications in mathematics and theoretical physics are discussed. This framework opens new avenues for the study of ordered structures, hierarchies, and parametric constraints in a more expressive way [1][2][3].