Compatible operations on commutative weak residuated lattices

Hernán Javier San Martín

doi:10.1007/s00012-015-0317-4

Abstract

Compatibility of functions is a classical topic in Universal Algebra related to the notion of affine completeness. In algebraic logic, it is concerned with the possibility of implicitly defining new connectives.

Highlights

The problem of adding connectives to extend a logic in a “natural” way has been broadly studied
Cignoli emphasizes the algebraic aspect of the problem through the notion of compatible function, which translates to the notion of compatible connective in intuitionistic logic
We study compatible functions in a new variety that includes the previous ones, providing a common framework to the results given in [5, 16]

Summary

Introduction

The problem of adding connectives to extend a logic in a “natural” way has been broadly studied. A generalized commutative residuated lattice, GCRL for short, is an algebra (A, ∧, ∨, ·, →, e) that satisfies the following conditions for every a, b, c ∈ A: 1. A commutative weak residuated lattice, CWRL for short, is a GCRL that satisfies the following conditions for every a, b, c ∈ A:. We write CWRL for the variety of CWRLs. A commutative residuated lattice (CRL for short) is an ordered algebraic structure (A, ∧, ∨, ·, →, e), where (A, ∧, ∨) is a lattice, (A, ·, e) is a commutative monoid, and → is a binary operation such that for every a, b, c ∈ A, the condition a · b ≤ c if and only if b ≤ a → c is satisfied. The choice of the conditions (R2)–(R7) is needed to characterize the compatible functions as a natural generalization of the case of the variety CRL This will become more clear in the development of this work.

Basic results

Convex subalgebras

Compatible functions

The minimum operator