P. Cousot & R. Cousot, A constructive characterization of the lattices of all retractions, pre-closure, quasi-closure and closure operators on a complete lattice

Patrick Cousot & Radhia Cousot.
A constructive characterization of the lattices of all retractions, pre-closure, quasi-closure and closure operators on a complete lattice.
In Portugaliæ Mathematica, 38(2):185--198, 1979.
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Abstract (from the introduction): We give a constructive characterization of the complete lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Our general approach is the following: in order to study the structure of the set G in (L -> L) of operators r on a complete lattice L satisfying a given axiom A, we show that r has property A if and only if it is a fixed point of some monotone operator F on the complete lattice (L -> L) proving that G is the set of fixed points of F. Then using Cousot & Cousot's contructive version of Tarski's lattice theoretical fixed point theorem, we constructively characterize the infimum, supremum, union and intersection of the complete lattice G which are defined by means of limits of ultimately stationary transfinite iterations sequences for F. Variants of this argment are used when F is a closure operator (in which case the constructive version of Tarski's fixpoint theorem amounts to Ward's theorem) or when the operators with property A are the postfiwed points of F or the common fixed points of two functionals. The reasoning is repeated when G is characterized by means of more than one axiom.

Bibliographic reference:

\bibitem{CousotCousot-PORTM79-38-2}
P{.} Cousot and R{.} Cousot.
\newblock A constructive characterization of the lattices of all
retractions, pre-closure, quasi-closure and closure operators on a
complete lattice.
\newblock \emph{Portugali\ae\ Mathematica}, 38(2):185--198, 1979.

@article{CousotCousot-PORTM79-38-2,
   author = {Cousot, P{.} and Cousot, R{.}},
   title =  {A constructive characterization of the lattices of all
             retractions, pre-closure, quasi-closure and closure operators 
             on a complete lattice},
   journal = {Portugali\ae\ Mathematica}, 
   volume = 38,
   number = 2,
   pages =  {185--198},
   year =   1979,
}

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