(Z(1), Z(2))-Complete Partially Ordered Sets and Their Representations by Q-Spaces


Demirci M.

APPLIED CATEGORICAL STRUCTURES, vol.21, no.6, pp.703-723, 2013 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 21 Issue: 6
  • Publication Date: 2013
  • Doi Number: 10.1007/s10485-012-9277-4
  • Journal Name: APPLIED CATEGORICAL STRUCTURES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.703-723
  • Keywords: Poset, (Z(1), Z(2))-complete poset, Q-space, Subset system, Sober, Spatial, CONTINUOUS-ALGEBRAS
  • Akdeniz University Affiliated: Yes

Abstract

The present paper proposes a general theory for -complete partially ordered sets (alias -join complete and -meet complete partially ordered sets) and their Stone-like representations. It is shown that for suitably chosen subset selections (i = 1,...,4) and , the category P of -complete partially ordered sets and -continuous (alias -join preserving and -meet preserving) functions forms a useful categorical framework for various order-theoretical constructs, and has a close connection with the category S of -spaces which are generalizations of topological spaces involving subset selections. In particular, this connection turns into a dual equivalence between the full subcategory P (s) of P of all -spatial objects and the full subcategory S (s) of S of all -sober objects. Here -spatiality and -sobriety extend usual notions of spatiality of locales and sobriety of topological spaces to the present approach, and their relations to -compact generation and -sobriety have also been pointed out in this paper.