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Duality for distributive and implicative semi-lattices

Guram Bezhanishvili, Ramon Jansana

Abstract

We develop a new duality for distributive and implicative meet semi-lattices. For distributive meet semi-lattices our duality generalizes Priestley's duality for distributive lattices and provides an improvement of Celani's duality. Our generalized Priestley spaces are similar to the ones constructed by Hansoul. Thus, one can view our duality for distributive meet semi-lattices as a completion of Hansoul's work. For implicative meet semi-lattices our duality generalizes Esakia's duality for Heyting algebras and provides an improvement of Vrancken-Mawet's and Celani's dualities. In the finite case it also yield's Köhler's duality. Thus, one can view our duality for implicative meet semi-lattices as a completion of Köhler's work. As a consequence, we also obtain a new duality for Heyting algebras, which is an alternative to the Esakia duality.

Duality for distributive and implicative semi-lattices

Abstract

We develop a new duality for distributive and implicative meet semi-lattices. For distributive meet semi-lattices our duality generalizes Priestley's duality for distributive lattices and provides an improvement of Celani's duality. Our generalized Priestley spaces are similar to the ones constructed by Hansoul. Thus, one can view our duality for distributive meet semi-lattices as a completion of Hansoul's work. For implicative meet semi-lattices our duality generalizes Esakia's duality for Heyting algebras and provides an improvement of Vrancken-Mawet's and Celani's dualities. In the finite case it also yield's Köhler's duality. Thus, one can view our duality for implicative meet semi-lattices as a completion of Köhler's work. As a consequence, we also obtain a new duality for Heyting algebras, which is an alternative to the Esakia duality.

Paper Structure

This paper contains 29 sections, 121 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Filters and ideals of meet semi-lattices
  4. Distributive envelopes, Frink ideals, and optimal filters
  5. Distributive envelopes
  6. Frink ideals
  7. Optimal filters
  8. Generalized Priestley spaces
  9. Generalized Esakia spaces
  10. Categorical equivalences
  11. The categories $\mathsf{GPS}$, $\mathsf{GPS}^{\mathsf{T}}$, and $\mathsf{GPS}^{\mathsf{F}}$
  12. The categories $\mathsf{GES}$, $\mathsf{GES}^{\mathsf{T}}$, and $\mathsf{GES}^{\mathsf{F}}$
  13. Functional morphisms
  14. Functional generalized Priestley morphisms
  15. Functional generalized Esakia morphisms
  16. ...and 14 more sections

Key Result

Proposition 2.1

The $\wedge$-reduct of an implicative meet semi-lattice is a distributive meet semi-lattice.

Theorems & Definitions (248)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Lemma 3.5: Prime Filter Lemma
  • Corollary 3.6
  • Proposition 3.7
  • ...and 238 more