Pointed semi-quantales and lattice-valued topological spaces


Demirci M.

FUZZY SETS AND SYSTEMS, vol.161, no.9, pp.1224-1241, 2010 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 161 Issue: 9
  • Publication Date: 2010
  • Doi Number: 10.1016/j.fss.2009.12.012
  • Journal Name: FUZZY SETS AND SYSTEMS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1224-1241
  • Keywords: Topology, Category theory, Many-valued topology, Lattice-valued topology, Quantales, Semi-quantales, SEPARATION AXIOMS, FUZZY, FOUNDATIONS, COMPACTNESS
  • Akdeniz University Affiliated: Yes

Abstract

In a way more general than variable-basis approach to lattice-valued topological spaces, the present paper introduces an alternative approach to lattice-valued topological spaces-direct product representation spaces extending the notion of quantal spaces in the sense of Mulvey and Pelletier to semi-quantales recently proposed by Rodabaugh. This paper aims to give an answer to the main question whether there exists a categorical connection, possibly a categorical equivalence, between direct product representation spaces and variable-basis lattice-valued topological spaces. Small sources in the category of semi-quantales which are called pointed semi-quantales can be identified with direct product representation spaces. For this reason, the main problem will be handled in terms of pointed semi-quantales. Generalized quasi-lattice-valued topological spaces extending variable-basis quasi-topological spaces into the present setting are introduced to be a suitable topological counterpart of pointed semi-quantales. To formalize and to solve the main problem, categories of pointed semi-quantales and of generalized quasi-lattice-valued topological spaces are constructed, and the relations between these categories, providing a satisfactory answer to the main problem, are established in this paper. (C) 2009 Elsevier B.V. All rights reserved.