Citation
Pierce, Richard Scott (1952) Homomorphisms of Function Lattices. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/VWRYVZ06. https://resolver.caltech.edu/CaltechTHESIS:09292017140233313
Abstract
This thesis is an algebraic study of systems of realvalued 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 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  1 January 1952 
Record Number:  CaltechTHESIS:09292017140233313 
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:09292017140233313 
DOI:  10.7907/VWRYVZ06 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  10470 
Collection:  CaltechTHESIS 
Deposited By:  Ben Perez 
Deposited On:  02 Oct 2017 15:05 
Last Modified:  21 Dec 2019 04:34 
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