Abstract
Celani and Jansana (Math Log Q 58(3):188–207, 2012) give an explicit description of the free implicative semilattice extension of a Hilbert algebra. In this paper, we give an alternative path conducing to this construction. Furthermore, following our procedure, we show that an adjunction can be obtained between the algebraic categories of Hilbert algebras with supremum and that of generalized Heyting algebras. Finally, in the last section, we describe a functor from the algebraic category of Hilbert algebras to that of generalized Heyting algebras, of possible independent interest.
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Acknowledgements
This work was supported by CONICET-Argentina (PIP 112-201501-00412). The authors thank the anonymous referee for the useful comments on the manuscript. In particular, for making us notice the connection of our construction with the logic-based canonical extension of Hilbert algebras. Hernán J. San Martín would also like to thank Ramón Jansana for useful discussions concerning the results of this work.
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Castiglioni, J.L., San Martín, H.J. Variations of the free implicative semilattice extension of a Hilbert algebra. Soft Comput 23, 4633–4641 (2019). https://doi.org/10.1007/s00500-018-3426-0
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DOI: https://doi.org/10.1007/s00500-018-3426-0