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

作者：Demirci Mustafa^*

来源：Applied Categorical Structures, 2013, 21(6): 703-723.

DOI：10.1007/s10485-012-9277-4

• Poset
• (Z(1), Z(2))-complete poset
• Q-space
• Subset system
• Sober
• Spatial

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.

• 出版日期2013-12