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On Varieties of Biresiduation Algebras

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Abstract

A biresiduation algebra is a 〈/,\,1〉-subreduct of an integral residuated lattice. These algebras arise as algebraic models of the implicational fragment of the Full Lambek Calculus with weakening. We axiomatize the quasi-variety B of biresiduation algebras using a construction for integral residuated lattices. We define a filter of a biresiduation algebra and show that the lattice of filters is isomorphic to the lattice of B-congruences and that these lattices are distributive. We give a finite basis of terms for generating filters and use this to characterize the subvarieties of B with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of B. The lattice of subvarieties of B is investigated; we show that there are precisely three finitely generated covers of the atom.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050 Johannesburg, South Africa

    C. J. van Alten

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  1. C. J. van Alten
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Correspondence to C. J. van Alten.

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Mathematics Subject Classification (2000): 03G25, 06F35, 06B10, 06B20

Dedicated to the memory of Willem Johannes Blok

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van Alten, C.J. On Varieties of Biresiduation Algebras. Stud Logica 83, 425–445 (2006). https://doi.org/10.1007/s11225-006-8312-6

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  • DOI: https://doi.org/10.1007/s11225-006-8312-6

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