This chapter is a core component of the book, dedicated to a detailed and in-depth investigation of the structure of hoops. Although the concept of ordinal sum was introduced in the previous chapters, its primary application in the representation of this algebraic structure, particularly irreducible hoops, is demonstrated in this chapter. The chapter delves into the structural analysis of hoops, focusing on specific classes and their representations across seven sections. We begin by exploring ordinal sums and \(\oplus \) -irreducible hoops, characterizing linear \(\oplus \) -irreducible hoops as Wajsberg hoops and demonstrating that linear Wajsberg hoops are either cancellative or bounded (bounded case implies that \(\textbf{H}\) is equivalent to an MV-algebra). A key theorem establishes that any linearly ordered hoop is an ordinal sum of Wajsberg hoops, leading to representations for BL-algebras and cancellative hoops.