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Basic Hoops: an Algebraic Study of Continuous t-norms

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Abstract

A continuous t-norm is a continuous map * from [0, 1]2 into [0, 1] such that \(\langle [0, 1], *, 1 \rangle\) 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 \(\langle [0, 1], *,\rightarrow, 1\rangle\) becomes a commutative naturally ordered residuated monoid, also called a hoop. The variety of basic hoops is precisely the variety generated by all algebras \(\langle [0, 1], *,\rightarrow, 1\rangle\) , 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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Via del Capitano 15, 53100, Siena, Italy

    P. Aglianò & F. Montagna

  2. Departamento de Matemática, FCUL, Campo Grande, Ed. C6, Piso 2, 1749-016, Lisboa, Portugal

    I. M. A. Ferreirim

Authors
  1. P. Aglianò
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  2. I. M. A. Ferreirim
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  3. F. Montagna
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Correspondence to P. Aglianò.

Additional information

Research partly supported by research projects Praxis 2/ 2.1/ MAT/73/94 “Álgebra e Matemáticas Discretas” and POCTI/0143/2003 “Fundamental and Applied Algebra”.

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Aglianò, P., Ferreirim, I.M.A. & Montagna, F. Basic Hoops: an Algebraic Study of Continuous t-norms. Stud Logica 87, 73–98 (2007). https://doi.org/10.1007/s11225-007-9078-1

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