Aglianò–Montagna type decomposition of linear pseudo hoops and its applications

We decompose every linear pseudo hoop as an Aglianò–Montagna type of ordinal sum of linear Wajsberg pseudo hoops which are either negative cones of linear ℓ-groups or intervals in linear unital ℓ-groups with strong unit. We apply the decomposition to present a new proof that every linear pseudo BL-algebra and consequently every representable pseudo BL-algebra is good. Moreover, we show that every maximal filter and every value of a linear pseudo hoop is normal, and every σ-complete linear pseudo hoop is commutative.