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Homomorphisms of Function Lattices


Pierce, Richard Scott (1952) Homomorphisms of Function Lattices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VWRY-VZ06.


This thesis is an algebraic study of systems of real-valued functions which are closed under the operations of pointwise meets and 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 are studied. This congruence is then applied to translation lattices, i.e., algebraic systems in which the two operations of meet and the addition of constants is defined. Results which are analogous to the isomorphism theorems of group theory are proved.

The second chapter contains the development of a representation theory for translation lattices. For this purpose, the concept of a normal lattice function is introduced. These functions are closely related to the normal functions on a topological space. It is shown that a translation lattice can always be mapped homomorphically onto a system of normal lattice functions. Uniqueness theorems are established for this representation.

Chapter three develops, briefly, a new method of constructing topological spaces from a complete Boolean algebra. In the final chapter, this construction is applied to show that a translation lattice can be represented as a translation lattice of continuous functions on a compact Hausdorff space. When suitable restrictions are imposed on the representation, this space -- called the characteristic space -- is uniquely determined. Finally, the relations between different representations by continuous functions are discussed. It is proved that the characteristic space, in an appropriate sense, is the minimal representation space.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics and Physics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Minor Option:Physics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Dilworth, Robert P.
Thesis Committee:
  • Unknown, Unknown
Defense Date:1 January 1952
Record Number:CaltechTHESIS:09292017-140233313
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10470
Deposited By: Benjamin Perez
Deposited On:02 Oct 2017 15:05
Last Modified:16 May 2023 21:30

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