We use the previous results to characterize finitely generated projective algebras in the two varieties, which turn out to be exactly the finitely presented algebras. From the point of view of the associated logics, via Ghilardi algebraic approach to unification problems [3], this implies that their unification type is (strongly) unitary: there is always a best solution to a unification problem, and it is represented algebraically by the identity homomorphism; this is in parallel to the case of product algebras and DLMV-algebras studied in [1]. The study of unification problems is strongly connected to the study of admissible rules (or, in the algebraic setting, admissible quasiequations); a rule is said to be admissible in a logic if every substitution that makes the premises a theorem of the logic, also makes the conclusion a theorem of the logic.

As a consequence of our results, we get that the logics associated to both product hoops and DLW-hoops are structurally complete, i.e. the admissibility of rules coincides with their derivability; using results in [2], we can actually conclude that the two logics are universally complete, that is, admissibility coincides with derivability also for multiple-conclusion rules.

References
[1] Aglianò, P., Ugolini, S.: Projectivity and unification in substructural logics of generalized rotations. Interna- tional Journal of Approximate Reasoning 153, 172–192, (2023).
[2] Aglianò, P., Ugolini, S.: Structural and universal completeness in algebra and logic. Submitted, (2023). arXiv:2309.14151
[3] Ghilardi S.: Unification through projectivity, J. Logic Comput. 7, 733–752, (1997).