A generalization of ideals in hoop algebras

This paper extends the study of ideals from bounded hoops to the more general setting of unbounded hoops, with a particular focus on Wajsberg hoops and totally ordered hoops. We introduce a new binary operation, ⊞, which plays a central role in this generalization, and investigate its fundamental properties. We provide various examples of ideals and show that every ideal I of a Wajsberg hoop H can be expressed as the union of a family of ideals of bounded Wajsberg hoops. As a result, each ideal is naturally associated with a filter on the Wajsberg hoop. We then examine ideals in ordinal sums of hoops and introduce a key property, denoted by (P*), which proves essential in characterizing ideals in totally ordered hoops.