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Minimal Varieties of Representable Commutative Residuated Lattices

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Abstract

We solve several open problems on the cardinality of atoms in the subvariety lattice of residuated lattices and FL-algebras [4, Problems 17–19, pp. 437]. Namely, we prove that the subvariety lattice of residuated lattices contains continuum many 4-potent commutative representable atoms. Analogous results apply also to atoms in the subvariety lattice of FL i -algebras and FL o -algebras. On the other hand, we show that the subvariety lattice of residuated lattices contains only five 3-potent commutative representable atoms and two integral commutative representable atoms. Inspired by the construction of atoms, we are also able to prove that the variety of integral commutative representable residuated lattices is generated by its 1-generated finite members.

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Authors and Affiliations

  1. Institute of Computer Science, Academy of Sciences of the Czech Republic, 182 07, Prague 8, Czech Republic

    Rostislav Horčík

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  1. Rostislav Horčík
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Correspondence to Rostislav Horčík.

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Horčík, R. Minimal Varieties of Representable Commutative Residuated Lattices. Stud Logica 100, 1063–1078 (2012). https://doi.org/10.1007/s11225-012-9456-1

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  • DOI: https://doi.org/10.1007/s11225-012-9456-1

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