This thesis is an algebraic study of systems of real-valuedfunctions which are closed under the operations of pointwise meetsand the addition of constants.
In the first chapter, a new kind of lattice congruence is defined in terms of lattice ideals. The properties of this congruence arestudied. This congruence is then applied to translation lattices, i.e.,algebraic systems in which the two operations of meet and the additionof constants is defined. Results which are analogous to the isomorphismtheorems of group theory are proved.
The second chapter contains the development of a representationtheory for translation lattices. For this purpose, the concept of a normal lattice function is introduced. These functions are closelyrelated to the normal functions on a topological space. It is shownthat a translation lattice can always be mapped homomorphically ontoa system of normal lattice functions. Uniqueness theorems areestablished for this representation.
Chapter three develops, briefly, a new method of constructingtopological spaces from a complete Boolean algebra. In the finalchapter, this construction is applied to show that a translation latticecan be represented as a translation lattice of continuous functions ona compact Hausdorff space. When suitable restrictions are imposed onthe representation, this space -- called the characteristic space --is uniquely determined. Finally, the relations between differentrepresentations by continuous functions are discussed. It is provedthat the characteristic space, in an appropriate sense, is the minimalrepresentation space.
