TRANSLATION LATTICES R. S . PIERCE University of Washington 1. INTRODUCTION During the past quarter century there has been a vigorous development of the algebraic aspects of analysis. The value of this work is consider- able. On the one hand, it has clarified the interrelations between some of the central theorems of classical analysis, and on the other hand, it has generalized and simplified the proofs of these theorems, thereby clear- ing the way for further advances. Much of the work on the algebra of analysis has aimed at establishing a suitable axiomatic basis for various general theories: Linear integral equations, harmonic analysis, potential theory, etc. A standard problem is that of characterizing the set of all continuous functions on a topological space in terms of some of their algebraic operations. In this paper, we are concerned with sets of real valued, continuous functions on a compact Hausdorff space with the operations of lattice meet and addition of a constant (translation). We will call these systems translation semi- lattices. Numerous examples of translation semi-lattices can be found in analysis: The upper (or lower) semi-continuous functions on a space, the super-harmonic functions on a domain of Euclidean space, certain families of convex sets in a Banach space, and many others. The -inspiration for this paper comes from two sources, [J] and [11]. Primarily it extends and generalizes the elegant work of Dilworth on normal functions. But in its abstract viewpoint it is more like Kaplansky!s paper on translation lattices. Our objective has been not only to develop a representation theory for abstract translation semi-lattices, but also to prove that there is a canonical representation theory and to develop some Received by the editors May 3, 1958.
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