<abstract xmlns="http://www.w3.org/1999/xhtml"><p> A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using examples of powerset theories in the category Set of sets and in the category of sets with similarity relations, it is proved that these theories can be used as ground theories for closure theories of powerset theories in these two categories. Moreover, it is proved that all these closure theories of powerset theories are topological constructs. A notion of a closure operator which preserves a canonical form of fuzzy objects in these categories is introduced, and it is proved that a closure theory of a powerset theory in the ground category Set is a coreflective subcategory of the closure theory of (Zadeh’s) powerset theory, which preserves canonical forms of fuzzy sets.</p></abstract>

A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using examples of powerset theories in the category Set of sets and in the category of sets with similarity relations, it is proved that these theories can be used as ground theories for closure theories of powerset theories in these two categories. Moreover, it is proved that all these closure theories of powerset theories are topological constructs. A notion of a closure operator which preserves a canonical form of fuzzy objects in these categories is introduced, and it is proved that a closure theory of a powerset theory in the ground category Set is a coreflective subcategory of the closure theory of (Zadeh’s) powerset theory, which preserves canonical forms of fuzzy sets.

Closure theories of powerset theories

University of Ostrava Institute for Research and Applications of Fuzzy Modeling Ostrava 1 CZECH REPUBLIC

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

{"article-title":"Closure theories of powerset theories"}

A notion of a closure theory of a powerset theory in a ground category is introduced as a generalization of a topology theory of a powerset theory. Using...

Closure theories of powerset theories

Published Online: Feb 19, 2016

Page range: 101 - 126

Received: Nov 21, 2015

DOI: https://doi.org/10.1515/tmmp-2015-0045

Keywords
closure theory, powerset theory, set with a similarity relation, topological construct

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Closure theories of powerset theories

Published Online: Feb 19, 2016

Page range: 101 - 126

Received: Nov 21, 2015

DOI: https://doi.org/10.1515/tmmp-2015-0045

Keywordsclosure theory, powerset theory, set with a similarity relation, topological construct

© 2016

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
closure theory, powerset theory, set with a similarity relation, topological construct