Pre-neighbourhood A-semi-lattices, neighbourhood lattices, and their associated continuous functions

This thesis is concerned with defining a topology-like structure on A-semi-lattices and examining the properties of the associated continuous functions. A general pre-neighbourhood structure, which satisfies the poset analogues of the neighbourhood systems of points in a topological space, is introduced on an A-semi-lattice, and is used to define open elements. However, without the complete atomic Boolean structure of P(X), the supremum of pre-neighbourhood open elements need not be open. Neighbourhood structures are introduced on arbitrary lattices to remove this deficiency.