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A Categorical Duality for Semilattices and Lattices

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Abstract

The main aim of this article is to develop a categorical duality between the category of semilattices with homomorphisms and a category of certain topological spaces with certain morphisms. The principal tool to achieve this goal is the notion of irreducible filter. Then, we apply this dual equivalence to obtain a topological duality for the category of bounded lattices and lattice homomorphism. We show that our topological dualities for semilattices and lattices are natural generalizations of the duality developed by Stone for distributive lattices through spectral spaces. Finally, we obtain directly the categorical equivalence between our topological spaces and those presented for Moshier and Jipsen (Algebra Univers 71(2):109–126, 2014).

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Acknowledgements

We greatly appreciate the comments and suggestions made by the Referee that helped to improve the paper. We also would like to thank the Editor for valuable comments.

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Authors and Affiliations

  1. Universidad Nacional del Centro de la Provincia de Buenos Aires, Tandil, Argentina

    Sergio A. Celani

  2. Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Santa Rosa, Argentina

    Luciano J. González

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  1. Sergio A. Celani
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  2. Luciano J. González
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Correspondence to Luciano J. González.

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Communicated by Jorge Picado.

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This research was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (Argentina) under the Grant PIP 112-20150-100412CO. The second author was also partially supported by Universidad Nacional de La Pampa under the Grant P.I. No 78 M, Res. 523/19.

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Celani, S.A., González, L.J. A Categorical Duality for Semilattices and Lattices. Appl Categor Struct 28, 853–875 (2020). https://doi.org/10.1007/s10485-020-09600-2

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  • DOI: https://doi.org/10.1007/s10485-020-09600-2

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