4 research outputs found
Poset products as relational models
We introduce a relational semantics based on poset products, and provide
sufficient conditions guaranteeing its soundness and completeness for various
substructural logics. We also demonstrate that our relational semantics unifies
and generalizes two semantics already appearing in the literature: Aguzzoli,
Bianchi, and Marra's temporal flow semantics for H\'ajek's basic logic, and
Lewis-Smith, Oliva, and Robinson's semantics for intuitionistic Lukasiewicz
logic. As a consequence of our general theory, we recover the soundness and
completeness results of these prior studies in a uniform fashion, and extend
them to infinitely-many other substructural logics
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
Gluing residuated lattices
We introduce and characterize various gluing constructions for residuated
lattices that intersect on a common subreduct, and which are subalgebras, or
appropriate subreducts, of the resulting structure. Starting from the 1-sum
construction (also known as ordinal sum for residuated structures), where
algebras that intersect only in the top element are glued together, we first
consider the gluing on a congruence filter, and then add a lattice ideal as
well. We characterize such constructions in terms of (possibly partial)
operators acting on (possibly partial) residuated structures. As particular
examples of gluing constructions, we obtain the non-commutative version of some
rotation constructions, and an interesting variety of semilinear residuated
lattices that are 2-potent. This study also serves as a first attempt toward
the study of amalgamation of non-commutative residuated lattices, by
constructing an amalgam in the special case where the common subalgebra in the
V-formation is either a special (congruence) filter or the union of a filter
and an ideal.Comment: This is a preprint. The final version of this work appears in Orde