Abstract
We study the pointed lattice subreducts of varieties of residuated lattices (RLs) and commutative residuated lattices (CRLs), i.e. lattice subreducts expanded by the constant \(\textsf{1}\) denoting the multiplicative unit. Given any positive universal class of pointed lattices \(\textsf{K}\) satisfying a certain equation, we describe the pointed lattice subreducts of semi-\(\textsf{K}\) and of pre-\(\textsf{K}\) RLs and CRLs. The quasivariety of semi-prime-pointed lattices generated by pointed lattices with a join prime constant \(\textsf{1}\) plays an important role here. In particular, the pointed lattice reducts of integral (semiconic) RLs and CRLs are precisely the integral (semiconic) semi-prime-pointed lattices. We also describe the pointed lattice subreducts of integral cancellative CRLs, proving in particular that every lattice is a subreduct of some integral cancellative CRL. This resolves an open problem about cancellative CRLs.
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21 August 2024
A Correction to this paper has been published: https://doi.org/10.1007/s11083-024-09679-5
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Acknowledgements
The author is grateful to an anonymous referee for their careful reading of the manuscript and a number of helpful suggestions, and to Nick Galatos and Peter Jipsen for their comments on an earlier draft of Section 4.
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Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was funded by the grant 2021 BP 00212 of the grant agency AGAUR of the Generalitat de Catalunya.
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Přenosil, A. Pointed Lattice Subreducts of Varieties of Residuated Lattices. Order (2024). https://doi.org/10.1007/s11083-024-09671-z
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DOI: https://doi.org/10.1007/s11083-024-09671-z