A Categorical Duality for Semilattices and Lattices
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A Categorical Duality for Semilattices and Lattices Sergio A. Celani1 · Luciano J. González2 Received: 21 February 2020 / Accepted: 27 May 2020 © Springer Nature B.V. 2020
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). Keywords Lattices · Semilattices · Duality · Filters Mathematics Subject Classification 06B15 · 06A12
1 Introduction The categorical dualities for ordered algebraic structures through topological spaces arose with the famous work of Stone [22] developing a categorical duality between the category of Boolean algebras with homomorphisms and the category of compact Hausdorff zerodimensional spaces (called Boolean spaces or Stone spaces) with continuous functions. Then, Stone himself extended this duality from Boolean algebras to bounded distributive
Communicated by Jorge Picado. 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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Luciano J. González [email protected] Sergio A. Celani [email protected]
1
Universidad Nacional del Centro de la Provincia de Buenos Aires, Tandil, Argentina
2
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de La Pampa, Santa Rosa, Argentina
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S. A. Celani, L. J. González
lattices through spectral spaces and spectral functions [13]. Some time later, Priestley developed another topological duality from a different approach for the category of distributive lattices employing compact totally order-disconnected spaces (called Priestley spaces) with continuous monotone functions. Both topological dualities for distributive lattices are very useful for the study of distributive lattices and they are also a powerful mathematical tool in the study of many non-classical logics having an algebraic semantics based on distributive lattices (see for instance [11,19]). From there, many generalizations of the topological dualities for distributive lattices following the Stone’s approach or the Priestley’s approach were obtained for several ordered algebraic structures having an adequate distributivity condition [1–3,5–7,9,11,12,14]. There are in the literature several categorical dualities for the category of arbitrary bo
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