U R. S. PIERCE u stand for the unit element* In a complete B.A., the least upper bound (respectively, greatest lower bound) of a set A is written either l.u.b. A (g.l.b.A), or V . a (A * a)*whichever is more convenient* a £ A a £ A If X is a topological space and Y is a subset of X, the closure of Y in X is written Y~ Also, the interior of Y, namely Y0"0 is designated Y°. It should be emphasized that the closure, interior and complement operations are always written exponentially in order from left to right a bar over a letter does not indicate topological closure. Often the closure operation is indicated without explicitly mentioning the topology. This is usually the case for subsets of a translation lattice which is always equipped with a natural metric topology. A subset A of a topological space X is called a regular open set if A"° » A. The collection of all such subsets form a complete Boolean algebra with the operations of finite set intersection and arbitrary joins given by V 6 e S A 6 - ^ 6 e s V " ° (see [2],p.177). This paper is based on the authorfs doctoral thesis written at the California Institute of Technology several years ago. The research was directed by Professor R. P. Dilworth who deserves much of the credit for its success. Acknowledgement is also due to Professor H. F. Bohnenblust for his proof of 5J below. For these contributions and even more, for their generous advice, inspiration and encouragement, I am pleased to be able to offer my warmest thanks. 2. HOMOMORPHISMS OF SEMI-IATTICES For future reference, we collect a few elementary definitions and facts concerning semi-lattices. Most of the results follow easily from the definitions. Complete proofs can be found in [12]• A semi-lattice S is a commutative semi-group in which every element is idempotent. The binary operation in S will be denoted by the symbol A # For f, g in S, define f g if f f A g, Then is a partial ordering of S such that a A b is the greatest lower bound of a and b in S.
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