On Varieties of Biresiduation Algebras

Studia Logica 83 (1/3):425 - 445 (2006)
A biresiduation algebra is a $\langle /,\setminus ,1\rangle $ -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 $\scr{B}$ 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 $\scr{B}$ -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 $\scr{B}$ with EDPC and also the discriminator varieties. A variety generated by a finite biresiduation algebra is shown to be a subvariety of $\scr{B}$ . The lattice of subvarieties of $\scr{B}$ is investigated; we show that there are precisely three finitely generated covers of the atom.
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DOI 10.2307/20016813
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