Achim Jung and Drew Moshier developed a Stone-type duality theory for bitopological spaces, amongst others, as a practical tool for solving a particular problem in the theory of stably compact spaces. By doing so they discovered that the duality of bitopological spaces and their algebraic counterparts, called d-frames, covers several of the known dualities. In this thesis we aim to take Jung's and Moshier's work as a starting point and fill in some of the missing aspects of the theory. In particular, we investigate basic categorical properties of d-frames, we give a Vietoris construction for d-frames which generalises the corresponding known Vietoris constructions for other categories, and we investigate the connection between bispaces and a paraconsistent logic and then develop a suitable (geometric) logic for d-frames.

This thesis is divided into two parts: Ordered spaces of Continuous Functions and the algebras associated with the topology of pointwise convergence of the associated construct, and Strictly completely regular bitopological spaces. The Motivation for part of the first part (Chapters 2, 3 and 4) comes from the recent study of function spaces for bitopological spaces in [44] and [45]. In these papers we see a clear generalisation of classical results in function spaces ( [14] and [55]) to bi-topological spaces. The well known definitions of the pointwise topology and the compact open topology in function spaces are generalized to bitopological spaces, and then familiar results such as Arens' theorem are generalised. We will use the same approach in chapters 2, 3 and 4 to formulate analogous definitions in the setting of ordered spaces. Well known results, including Arens' theorem, are also generalised to ordered spaces. In these chapters we will also compare function spaces in the category of topological spaces and continuous functions, the category of bi topological spaces and bicontinuous functions, and the category of ordered topological spaces and continuous order-preserving functions. This work has resulted in the publication of [30] and [31]. Continuing our study of Function Spaces, we oonsider in Chapters 5 and 6 some Categorical aspects of the construction, motivated by a series of papers which includes [39], [40], [41] and [50]. In these papers the Eilenberg-Moore Category of algebras of the monad induced by the Hom-functor on the categories of sets and categories of topological spaces are classified. Instead of looking at the whole product topology we will restrict ourselves to the pointwise topology and give examples of the EilenbergMoore Algebras arising from this restriction. We first start by way of motivation, with the discussion of the monad when the range space is the real line with the usual topology. We then restrict our range space to the two point Sierpinski space, with the aim of discovering a topological analogue of the well known characterization of Frames as the Eilenberg-Moore Category of algebras associated with the Hom-F\mctor of maps into the Sierpinski space [11]. In this case the order structure features prominently, resulting in the category Frames with a special property called "balanced" and Frame homomorphisms as the Eilenberg-Moore category of M-algebras. This has resulted in [34]. The Motivation for the second part comes from [20] and [15]. In [20], J. D. Lawson introduced the notion of strict complete regularity in ordered spaces. A detailed study of this notion was done by H-P. A. Kiinzi in [15]. We shall introduce an analogous notion for bitopological spaces, and then shall also compare the two notions in the categories of bi topological spaces and bicontinuous functions, and of ordered topological spaces and continuous order-preserving functions via the natural functors considered in the previous chapters. We further study the Stone-Cech bicompactification and Stone-Cech ordered compactification in the two categories. This has resulted in [32] and [33] / Mathematical Sciences / D. Phil. (Mathematics)