Double Approximation and Complete Lattices

2011 ◽  
Vol 111 (1) ◽  
pp. 1-14
Author(s):  
Taichi Haruna ◽  
Yukio-Pegio Gunji
Keyword(s):  
Complete Lattices ◽  
Double Approximation

Q-orders with no unit on Q

2021 ◽  
pp. 1-8
Author(s):  
Cheng-Jie Zhou ◽  
Wei Yao
Keyword(s):  
Ordered Sets ◽  
Complete Lattices ◽  
Definition Of ◽  
Self Adaptive

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.

scholarly journals On complete objects in the category of T0 closure spaces

2003 ◽  
Vol 4 (1) ◽  
pp. 25 ◽  
Author(s):  
D. Deses ◽  
Eraldo Giuli ◽  
E. Lowen-Colebunders
Keyword(s):  
General Theory ◽  
Complete Lattices ◽  
Closure Spaces

<p>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 Cl<sub>0</sub> of T<sub>0</sub> 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 T<sub>0</sub> closure spaces is dually equivalent to the category of complete lattices with maps preserving the top and arbitrary joins.</p>

scholarly journals Usage of Invariants for Symbolic Verification of Requirements

10.29007/2nr2 ◽  
2018 ◽  
Author(s):  
Alexander Letichevsky ◽  
Alexander Godlevsky ◽  
Anton Guba ◽  
Alexander Kolchin ◽  
Oleksandr Letychevskyi ◽  
...  
Keyword(s):  
Iterative Method ◽  
Reactive Systems ◽  
Consistency Properties ◽  
Double Approximation ◽  
Symbolic Verification

The paper presents the usage of invariants for symbolic verification of requirements for reactive systems. It includes checking of safety, incompleteness, liveness, consistency properties, and livelock detection. The paper describes the iterative method of double approximation and the method of undetermined coefficients for invariants generation. Benefits, disadvantages, and comparison of this technique with existing methods are considered. The paper is illustrated by examples of invariants technique usage for symbolic verification.

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