Algebraic results on universal quantifiers in monoidal t-norm based logic

TL;DR

This paper introduces and studies UMTL-algebras, an extension of MTL-algebras with a universal quantifier, and establishes a modal logic complete with respect to these algebras.

Contribution

It extends MTL-algebras by adding a universal quantifier, characterizes subclasses, and develops a modal logic complete with respect to UMTL-algebras.

Findings

UMTL-algebras generalize existing algebraic structures

Modal monoidal t-norm based logic is complete w.r.t. UMTL-algebras

Semilinearity characterized by specific conditions

Abstract

In this paper, we enlarge the language of MTL-algebras by a unary operation $\forall$ equationally described so as to abstract algebraic properties of the universal quantifier "for any" in its original meaning. The resulting class of algebras will be called \emph{MTL-algebras with universal quantifiers} (UMTL-algebras for short). After discussing some basic algebraic properties of UMTL-algebras, we start a systematic study of the main subclasses of UMTL-algebras, some of which constitute well known algebras: UMV-algebras and monadic Boolean algebra. Then we give some characterizations of representable, simple, semsimple UMTL-algebras, and obtain some representations of UMTL-algebras. Finally, we establish modal monoidal t-norm based logic and prove that is completeness with respect to the variety of UMTL-algebras, and then obtain that a necessary and sufficient condition for the modal monoidal t-norm based logic to be semilinear.