Duality for distributive and implicative semi-lattices

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 K\"ohler's duality. Thus, one can view our duality for implicative meet semi-lattices as a completion of K\"ohler's work. As a consequence, we also obtain a new duality for Heyting algebras, which is an alternative to the Esakia duality.