A space of pseudoquotients can be described as a space of either single term quotients (the injective case) or the quotient of sequences (the non-injective case) where the parent sets for the numerator and the denominator satisfy particular conditions. The first part of this project is concerned with the minimal of conditions required to have a well-defined set of pseudoquotients. We continue by adding more structure to our sets and discuss the effect on the resultant pseudoquotient. Pseudoquotients can be thought of as extensions of the parent set for the numerator since they include a natural embedding of that set. We answer some questions about the extension properties. One family of these questions involves assuming a structure (algebraic or topological) on a set and asking if the set of pseudoquotients generated has the same structure. A second family of questions looks at maps between two sets and asks if there is an extension of that map between the corresponding pseudoquotients? If so, do the properties of the original map survive the extension? The result of our investigations on the abstract setting will be compared with some well-known spaces of pseudoquotients and Boehmians (a particular case of non-injective pseudoquotients). We will show that the conditions discussed in the first part are satisfied and we will use that to reach conclusions about our extension spaces and the extension maps. The Fourier transform is one of the maps that we will continuously revisit and discuss. Finally many spaces of Boehmians have been introduced where the initial set is a particular class of functions on either locally compact groups R and or a compact group such as a sphere. The natural question is, can we generalize the construction to any locally compact group. In some previous work such construction is discussed, however here we go further; we use characters and define the Fourier transform of integrable and square integrable Boehmians on a locally compact group. Then we discuss the properties of such transform.

It is shown that Boehmians defined over open sets of JRN constitute a sheaf. In particular, it is shown that such Boehmians satisfy the gluing property of sheaves over topological spaces.