Closed frame homomorphisms

Abstract

<p>This thesis is a systematic study of closed frame homomorphisms, which can be viewed as a natural generalization of the classical closed continuous mappings of topological spaces. Following the features of frame theory, we attempt to prove our results constructively. Various aspects of closed homomorphisms are investigated in relation to certain categorical colimits, including coequalizers, coproducts and pushouts. Another main topic is the study of perfect homomorphisms. Useful characterizations are obtained for perfect homomorphisms between regular continuous frames and between completely regular frames. The injectives in the category of completely regular frames are analysed. A condition equivalent to the Sikorski Theorem (injective = complete, for Boolean algebra) is established. As a fundamental part of the whole work, the structure of binary coproducts of frames is studied in a constructive context. Finally, the basic theory of connected congruences is developed and applied to study local connectedness of frames.</p>