The Structure of Generalized BI-algebras and Weakening Relation Algebras

Generalized bunched implication algebras (GBI-algebras) are defined as residuated lattices with a Heyting implication, and are positioned between Boolean algebras with operators and lattices with operators. We characterize congruences on GBI-algebras by filters that are closed under Gumm–Ursini terms, and for involutive GBI-algebras these terms simplify to a dual version of the congruence term for relation algebras together with two more terms. We prove that representable weakening relation algebras form a variety of cyclic involutive GBI-algebras, denoted by RWkRA, containing the variety of representable relation algebras. We describe a double-division conucleus construction on residuated lattices and on (cyclic involutive) GBI-algebras and show that it generalizes Comer’s double coset construction for relation algebras. Also, we explore how the double-division conucleus construction interacts with other class operators and in particular with variety generation. We focus on the fact that it preserves a special discriminator term, thus yielding interesting discriminator varieties of GBI-algebras, including RWkRA. To illustrate the generality of the variety of weakening relation algebras, we prove that all distributive lattice-ordered pregroups and hence all lattice-ordered groups embed, as residuated lattices, into representable weakening relation algebras on chains. Moreover, every representable weakening relation algebra is embedded in the algebra of all residuated maps on a doubly-algebraic distributive lattice. We give a number of other instructive examples that show how the double-division conucleus illuminates the structure of distributive involutive residuated lattices and GBI-algebras

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 (BL-algebras), we 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 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 2n style= position: relative; tabindex= 0 id= MathJax-Element-1-Frame \u3e2n 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 it