Bounded distributive lattices, priestley spaces and dual categorical equivalences between them

In this thesis, two main topics are emphasized based on category theory. These issues are discussed under the sections "Priestley Duality" and "An Extension of Priestley Duality". Priestley duality states that the category of bounded and distributive lattices and the category of Priestley spaces are equivalent categories. To show this, a pair of categorical functions called equivalence functors are used. These functors are clearly expressed in the thesis. In this way, in many areas, such as Stone algebras, Kleene algebras, Ockham algebras, De Morgan algebras, Wajsberg algebras; contributing to the development of the theory of lattice theory is intended to contribute to mathematics. The work of Hilary Priestley, especially in the early 1970s, is well known in this field. In the section of "An Extension of Priestley Duality" a generalization of Priestley duality is formulated in the same way. The aim of this generalization is that, Priestley duality involves too many conditions. Because in mathematics, such structures may not always be found. In doing so, we have extended the lattice homomorphisms to join-homomorpisms and the continuous and order-preserving functions to Priestley relations. The structure of Priestley duality is not the only reason to give this generalisation. By making use of this duality, a duality for an extended category of Boolean algebras, the so-called Halmos-Wright duality, has been formulated. In addition, the renowned Stone duality, that is stating a dual equivalence between the category of Boolean algebras and the category of Stone spaces, is obtained as a consequence of Priestley duality.