Peter Jipsen From Semirings to Residuated Kleene Lattices
نویسنده
چکیده
We consider various classes of algebras obtained by expanding idempotent semirings with meet, residuals and Kleene-∗. An investigation of congruence properties (epermutability, e-regularity, congruence distributivity) is followed by a section on algebraic Gentzen systems for proving inequalities in idempotent semirings, in residuated lattices, and in (residuated) Kleene lattices (with cut). Finally we define (one-sorted) residuated Kleene lattices with tests to complement two-sorted Kleene algebras with tests.
منابع مشابه
An Overview of Residuated Kleene Algebras and Lattices
1. Residuated Lattices with iteration 2. Background: Semirings and Kleene algebras 3. A Gentzen system for Residuated Kleene Lattices and some reducts 4. Interpreting Kleene algebras with tests 1. Residuated Lattices with iteration This talk is mostly about Residuated Kleene Lattices, which are defined as noncommutative residuated 0,1-lattices expanded with a unary operation * that satisfies x ...
متن کاملRelation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras
This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idem...
متن کاملUntyping Typed Algebras and Colouring Cyclic Linear Logic
We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic li...
متن کاملUntyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic
We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic li...
متن کاملA Survey of Generalized Basic Logic Al- gebras
Petr Hájek identified the logic BL, that was later shown to be the logic of continuous t-norms on the unit interval, and defined the corresponding algebraic models, BL-algebras, in the context of residuated lattices. The defining characteristics of BL-algebras are representability and divisibility. In this short note we survey recent developments in the study of divisible residuated lattices an...
متن کامل