Abstract
We expand \(\hbox {FL}_{ew}\) with a unary connective whose algebraic counterpart is the operation that gives the greatest complemented element below a given argument. We prove that the expanded logic is conservative and has the Finite Model Property. We also prove that the corresponding expansion of the class of residuated lattices is an equational class.
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Notes
Actually, this could be generalized in the following sense. In Noguera (2007), Noguera proves that the variety generated by simple n-contractive MTL chains is the variety of \(S_n\)-MTL algebras, i.e., MTL algebras satisfying the law \(x \vee \lnot x^{n-1}= 1\). Therefore, any variety of n-contractive MTL algebras that are not \(S_n\)-MTL has a chain with a proper filter. In particular, this is the case for the varieties of WNM and NM algebras, since they are 3-contractive and not \(S_3\)-MTL.
Notice that not all instances of De Morgan laws are valid in the variety of MTL algebras; for instance, the equations \(\lnot (x \wedge y) \approx \lnot x \vee \lnot y\) and \(\lnot (x \vee y) \approx \lnot x \wedge \lnot y\) are valid, but \(x \wedge y \approx \lnot (\lnot x \vee \lnot y)\) is not.
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Acknowledgments
The authors are thankful to an anonymous reviewer for his/her comments that have helped to improve the final layout of this paper. The authors have been funded by the EU H2020-MSCA-RISE-2015 Project 689176–SYSMICS. Esteva and Godo have been also funded by the FEDER/MINECO Spanish project TIN2015-71799-C2-1-P.
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Communicated by A. Di Nola, D. Mundici, C. Toffalori, A. Ursini.
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Ertola-Biraben, R.C., Esteva, F. & Godo, L. Expanding \(\hbox {FL}_{ew}\) with a Boolean connective. Soft Comput 21, 97–111 (2017). https://doi.org/10.1007/s00500-016-2275-y
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DOI: https://doi.org/10.1007/s00500-016-2275-y