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

Jun Tao Wang

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arXiv · 1910.04015

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

Jun Tao Wang

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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.

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BibTeX RIS

Jun Tao Wang. 2019-10-09. Algebraic results on universal quantifiers in monoidal t-norm based logic. https://arxiv.org/abs/1910.04015

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