On generalized hoops, homomorphic images of residuated lattices, and (G)BL-algebras

Right-residuated binars and right-divisible residuated binars are defined as precursors of generalized hoops, followed by some results and open problems about these partially ordered algebras. Next we 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 (BLalgebras), we give an effective description of the HSposet 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 between complete perfect Heyting algebras and partially ordered sets. We 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 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 between some substructural logics and classical modal logic. Finally we 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. 1 Residuated binars and generalized hoops We begin by considering structures with a simpler signature than residuated lattices. The aim of this section is to focus on the right-divisibility axiom in the setting Faculty of Mathematics, School of Computational Sciences and Center of Excellence in Computation, Algebra and Topology Chapman University, Orange, CA 92866, USA E-mail: [email protected] of right-residuated structures, and without further assumptions such as associativity or commutativity. A right-residuated binar is of the form (A,≤, ·, /) where (A,≤) is a partially ordered set, · is a binary operation on A and / is its right residual. This means that for all x, y, z ∈ A