Representation of Complete Lattices by Sets

For a usual commutative quantale Q (does not necessarily have a unit), we propose a definition of Q-ordered sets by introducing a kind of self-adaptive self-reflexivity. We study their completeness and the related Q-modules of complete lattices. The main result is that, the complete Q-ordered sets and the Q-modules of complete lattices are categorical isomorphic.

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Connectivity on Complete Lattices

Mathematical Morphology and its Applications to Image and Signal Processing - Computational Imaging and Vision ◽

10.1007/978-1-4613-0469-2_11 ◽

1996 ◽

pp. 81-96 ◽

Cited By ~ 7

Author(s):

Jean Serra

Keyword(s):

Complete Lattices

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On complete objects in the category of T0 closure spaces

Applied General Topology ◽

10.4995/agt.2003.2007 ◽

2003 ◽

Vol 4 (1) ◽

pp. 25 ◽

Cited By ~ 6

Author(s):

D. Deses ◽

Eraldo Giuli ◽

E. Lowen-Colebunders

Keyword(s):

General Theory ◽

Complete Lattices ◽

Closure Spaces

In this paper we present an example in the setting of closure spaces that fits in the general theory on “complete objects” as developed by G. C. L. Brümmer and E. Giuli. For V the class of epimorphic embeddings in the construct Cl0 of T0 closure spaces we prove that the class of V-injective objects is the unique firmly V-reflective subconstruct of Cl0. We present an internal characterization of the Vinjective objects as “complete” ones and it turns out that this notion of completeness, when applied to the topological setting is much stronger than sobriety. An external characterization of completeness is obtained making use of the well known natural correspondence of closures with complete lattices. We prove that the construct of complete T0 closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins.

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