Distributive Lattices with a Generalized Implication: Topological Duality

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• 作者：1. Universidad Nacional de San Juan ; San Juan ; Argentina2. CONICET and Departamento de Matemáticas ; Universidad Nacional del Centro ; Tandil ; Argentina3. Dept. Lògica ; Història i Filsofia de la Ciència ; Universitat de Barcelona ; Montalegre 6 ; 08001 Barcleona ; Spain
• 关键词：Generalized implication – Quasi ; modal operator – Annihilator – Weakly Heyting algebras – Priestley’s duality
• 刊名：Order
• 出版年：2011
• 出版时间：July 2011
• 年：2011
• 卷：28
• 期：2
• 页码：227-249
• 全文大小：414.9 KB
• 参考文献：1. Castro, J., Celani, S.A.: Quasi-modal lattices. Order 21, 107–129 (2004)
  2. Celani, S.A.: Quasi-modal algebras. Math. Bohem. 126, 721–736 (2001)
  3. Celani, S.A., Jansana R.: A closer look at some subintuitionistic logics. Notre Dame J. Form. Log. 42, 225–255 (2003)
  4. Celani, S.A., Jansana R.: Bounded distributive lattices with strict implication. Math. Log. Q. 51, 219–246 (2005)
  5. Mandelker, M.: Relative annihilators in lattices. Duke Math. J. 37, 377–386 (1970)
  6. Priestley, H.A.: Representation of distributive lattices by means of ordered Stone spaces. Bull. Lond. Math. Soc. 2, 186–190 (1970)
  7. Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. Lond. Math. Soc. 3, 507–530 (1972)
• 作者单位：http://www.springerlink.com/content/728p40285184265u/
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Convex and Discrete Geometry
  Geometry
  Theory of Computation
  
• 出版者：Springer Netherlands
• ISSN：1572-9273

文摘

In this paper we introduce the notion of generalized implication for lattices, as a binary function ? that maps every pair of elements of a lattice to an ideal. We prove that a bounded lattice A is distributive if and only if there exists a generalized implication ? defined in A satisfying certain conditions, and we study the class of bounded distributive lattices A endowed with a generalized implication as a common abstraction of the notions of annihilator (Mandelker, Duke Math J 37:377–386, 1970), Quasi-modal algebras (Celani, Math Bohem 126:721–736, 2001), and weakly Heyting algebras (Celani and Jansana, Math Log Q 51:219–246, 2005). We introduce the suitable notions of morphisms in order to obtain a category, as well as the corresponding notion of congruence. We develop a Priestley style topological duality for the bounded distributive lattices with a generalized implication. This duality generalizes the duality given in Celani and Jansana (Math Log Q 51:219–246, 2005) for weakly Heyting algebras and the duality given in Celani (Math Bohem 126:721–736, 2001) for Quasi-modal algebras.