Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, follod by some results and open problems about these partially ordered algebras. Next  show that all complete homomorphic images of a complete residuated lattice A can be constructed easily on certain definable subsets of A. Applying these observations to the algebras of Hajek’s basic logic (BL-algebras),  give an effective description of the HS-poset of finite subdirectly irreducible BL-algebras. The lattice of finitely generated BL-varieties can be obtained from this HS-poset by constructing the lattice of downward closed sets. These results are extended to bounded generalized BL-algebras using poset products and the duality beten complete perfect Heyting algebras and partially ordered sets.  also prove that the number of finite generalized BL-algebras with n join-irreducible elements is, up to isomorphism, the same as the number of preorders on an n-element set, hence the same as the number of closure algebras (i.e., S4-modal algebras) with \(2^{n}\) elements. This result gives rise to a faithful functor from the category of finite GBL-algebras to the category of finite closure algebras that is full on objects, providing a novel connection beten some substructural logics and classical modal logic. Finally,  show how generic satisfaction modulo theories solvers (SMT-solvers) can be used to obtain practical decision procedures for propositional basic logic and many of its extensions.
['Peter Jipsen']
I am writing a research paper on a idea On generalized hoops, homomorphic images of residuated lattices, and (G)BL-algebras
[{"from": "human", "value": "I am writing a research paper on a idea On generalized hoops, homomorphic images of residuated lattices, and (G)BL-algebras"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, follod by some results and open problems about these partially ordered algebras. Next  show that all complete homomorphic images of a complete residuated lattice A can be constructed easily on certain definable subsets of A. Applying these observations to the algebras of Hajek\u2019s basic logic (BL-algebras),  give an effective description of the HS-poset of finite subdirectly irreducible BL-algebras. The lattice of finitely generated BL-varieties can be obtained from this HS-poset by constructing the lattice of downward closed sets. These results are extended to bounded generalized BL-algebras using poset products and the duality beten complete perfect Heyting algebras and partially ordered sets.  also prove that the number of finite generalized BL-algebras with n join-irreducible elements is, up to isomorphism, the same as the number of preorders on an n-element set, hence the same as the number of closure algebras (i.e., S4-modal algebras) with \\(2^{n}\\) elements. This result gives rise to a faithful functor from the category of finite GBL-algebras to the category of finite closure algebras that is full on objects, providing a novel connection beten some substructural logics and classical modal logic. Finally,  show how generic satisfaction modulo theories solvers (SMT-solvers) can be used to obtain practical decision procedures for propositional basic logic and many of its extensions."}]