Studia Logica 87 (1):73 - 98 (2007)

Abstract
A continuoxis t- norm is a continuous map * from [0, 1]² into [0,1] such that ([ 0,1], *, 1) is a commutative totally ordered monoid. Since the natural ordering on [0,1] is a complete lattice ordering, each continuous t-norm induces naturally a residuation → and ([ 0,1], *, →, 1) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras ([ 0,1], *, →, 1), where * is a continuous t-norm. In this paper we investigate the structure of the variety of basic hoops and some of its subvarieties. In particular we provide a complete description of the finite subdirectly irreducible basic hoops, and we show that the variety of basic hoops is generated as a quasivariety by its finite algebras. We extend these results to Hájek's BL-algebras, and we give an alternative proof of the fact that the variety of BL-algebras is generated by all algebras arising from continuous t-norms on [ 0,1] and their residua. The last part of the paper is devoted to the investigation of the subreducts of BL- algebras, of Gödel algebras and of product algebras
Keywords Philosophy   Computational Linguistics   Mathematical Logic and Foundations   Logic
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DOI 10.1007/s11225-007-9078-1
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References found in this work BETA

A Propositional Calculus with Denumerable Matrix.Michael Dummett - 1959 - Journal of Symbolic Logic 24 (2):97-106.
A Complete Many-Valued Logic with Product-Conjunction.Petr Hájek, Lluis Godo & Francesc Esteva - 1996 - Archive for Mathematical Logic 35 (3):191-208.
The Separation Theorem of Intuitionist Propositional Calculus.Alfred Horn - 1962 - Journal of Symbolic Logic 27 (4):391-399.
Equational Classes of Relative Stone Algebras.T. Hecht & Tibor Katriňák - 1972 - Notre Dame Journal of Formal Logic 13 (2):248-254.

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Citations of this work BETA

Structural Completeness in Fuzzy Logics.Petr Cintula & George Metcalfe - 2009 - Notre Dame Journal of Formal Logic 50 (2):153-182.
Formal Systems of Fuzzy Logic and Their Fragments.Petr Cintula, Petr Hájek & Rostislav Horčík - 2007 - Annals of Pure and Applied Logic 150 (1-3):40-65.
Mathematical Fuzzy Logics.Siegfried Gottwald - 2008 - Bulletin of Symbolic Logic 14 (2):210-239.
Varieties of BL-Algebras III: Splitting Algebras.Paolo Aglianó - 2019 - Studia Logica 107 (6):1235-1259.

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