Nodes of pseudo-hoops

We investigate nodes (elements that are comparable with all other elements) and their role in the structural analysis of pseudo-hoops. First, we study idempotent nodes and present a complete representation of pseudo-hoops with finitely many idempotent nodes, based on their decomposition into ordinal sums of pseudo-hoops. Next, we examine ordinal sum irreducible pseudo-hoops and prove that every Wajsberg pseudo-hoop with a non-trivial node is linearly ordered. Moreover, we show that the set of nodes Nod( E) of the ordinal sum of any family of Wajsberg pseudo-hoops forms a subalgebra. We then use the set of regular elements of a bounded representable pseudo-hoop E and its relationship with Nod( E) to determine whether E is ordinal sum reducible or irreducible. As a consequence, we establish that every finite basic pseudo-hoop can be expressed as an ordinal sum of a finite family E₁,… , Eₙ for some n∈ N, where E₁,… , Eₙ₋₁ are linear Wajsberg pseudo-hoops and Eₙ is ordinal sum irreducible.