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Finite Distributive Semilattices

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Abstract

The present article aims to develop a categorical duality for the category of finite distributive join-semilattices and \(\wedge \)-homomorphisms (maps that preserve the joins and the meets, when they exist). This dual equivalence is a generalization of the famous categorical duality given by Birkhoff for finite distributive lattices. Moreover, we show that every finite distributive semilattice is a Hilbert algebra with supremum. We obtain some applications from the dual equivalence. We provide a dual description of the 1–1 and onto \(\wedge \)-homomorphisms, and we obtain a dual characterization of some subalgebras. Finally, we present a representation for the class of finite semi-boolean algebras.

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Notes

  1. Notice that we are allowing that the empty set is an ideal.

  2. This result is a direct generalization from the lattice case, and it was proved by Dr. Ismael Calomino.

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Acknowledgements

I am very grateful to the anonymous reviewer for his/her comments and suggestions that helped me to improve this article.

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Authors and Affiliations

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

    Luciano J. González

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  1. 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 work was partially supported by FONCyT-ANPCyT (Argentina) under the Grant PICT-2019-00674, by Universidad Nacional de La Pampa under the Grant P.I. No 78M, Res. 523/19, and by FONCyT-ANPCyT (Argentina) under the Grant PICT-2019-00882.

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González, L.J. Finite Distributive Semilattices. Appl Categor Struct 30, 641–658 (2022). https://doi.org/10.1007/s10485-021-09669-3

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