P. Cousot and R. Cousot.A constructive characterization of the lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Portugal. Math., 38(2):185-- 198, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... is satisfied for each non empty set X A (f is also called complete join morphism) An upper closure operator on a partially ordered set hA; i is a function ae : A A that is idempotent, i.e. ae(ae(c) ae(c) extensive, i.e. c ae(c) and monotonic (more on closure operators can be found in [23]) Let (L; be a non empty complete lattice. Let f : L L be a function. The ordinal powers of f are defined as follows for x 2 L: f 0(x) x f ff(x) f(f (ff Gamma 1) x) for every successor ordinal ff; and f ff(x) ffi ff f ffi(x) for every limit ordinal ff. The ....

P. Cousot and R. Cousot. A constructive characterization of the lattices of all retracts, pre-closure, quasi-closure and closure operators on a complete lattice. Portugaliae Mathematica, 38(2):185--198, 1979.

.... is satisfied for each non empty set X A (f is also called complete join morphism) An upper closure operator on a partially ordered set hA; i is a function ae : A A that is idempotent, i.e. ae(ae(c) ae(c) extensive, i.e. c ae(c) and monotonic (more on closure operators can be found in [23]) Let (L; be a non empty complete lattice. Let f : L L be a function. The ordinal powers of f are defined as follows for x 2 L: f 0(x) x f ff(x) f(f (ff Gamma 1) x) for every successor ordinal ff; and f ff(x) ffi ff f ffi (x) for every limit ordinal ff. The ....

P. Cousot and R. Cousot. A constructive characterization of the lattices of all retracts, pre-closure, quasi-closure and closure operators on a complete lattice. Portugaliae Mathematica, 38(2):185--198, 1979.

....if for all P 2 Program, ff(lfp(T P ) A lfp(T ] P ) This soundness condition can be more easily verified by checking whether for all P 2 Program, ff ffi T P A T ] P ffi ff, or, equivalently, ff ffi T P ffi fl C T ] P . In fact, a standard well known result of abstract interpretation (cf. [Cousot and Cousot 1979b, Theorem 7.1.0.2] ensures that this condition on T implies the soundness for S ] We then say that S ] is a fully sound abstraction of S if for all P 2 Program, ff ffi T P A T ] P ffi ff. Often, we will slightly abuse on terminology, by saying that some abstract semantic function T ] P is (fully) ....

.... on C for f , namely: Delta(C ; f ) fae 2 uco(C ) j ae(lfp(f ) lfp(ae ffi f )g: Also, if j 2 uco(C ) then Gamma j (C ; f ) and Delta j (C ; f ) are the set of closures on hj(C ) C i that are, respectively, fully complete and complete (for f ) It is helpful to recall (see e.g. [Cousot 1978, Theorem 4.2.0.4.7] for a detailed proof) that if j 2 uco(C ) then uco(j(C ) is isomorphic to the principal filter of uco(C ) generated by j, i.e. huco(j(C ) vi = h j; vi, and therefore ae 2 uco(j(C ) iff ae 2 uco(C ) and ae v j. By this remark, we get that Gamma j (C ; f ) Gamma(C ; f ) j ....

Cousot, P. and Cousot, R. 1979a. A constructive characterization of the lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Portug. Math. 38, 2, 185--198.

.... operator such that for all e 2 L: f(f(e) f(e) idempotent) An upper closure operator (uco) on L is a retraction ae such that 8e 2 L: e ae(e) extensive) a lower closure operator (lco) on L is a retraction ae such that 8e 2 L: ae(e) e (reductive) More on closure operators can be found in [9]. Let f : L L be a monotonic operator on the complete lattice h L; i. The upper ordinal powers of f are defined as follows: f 0 = f j = f(f (j Gamma 1) if j is a successor ordinal f = j f j if is a limit ordinal. The first limit ordinal equipotent with the ....

P. Cousot and R. Cousot. A constructive characterization of the lattices of all retracts, pre-closure, quasi-closure and closure operators on a complete lattice. Portugaliae Mathematica, 38(2):185--198, 1979.

....ordered set #L, ## is a monotonic and idempotent operator. An upper closure operator (uco) on L is a retraction # such that #x # L.x # #(x) extensive) a lowerclosure operator (lco) on L is a retraction # such that #x # L. #(x) # x (reductive) More on closure operators can be found in [CC79a, Mor60]. Let #L, #, #, #, #, ## be a nonempty complete lattice, and f : L # L. The upper ordinal powers of f are defined as follows: f # 0(X) X f # #(X) f(f # (# 1) X) for every successor ordinal #; and f # #(X) # # # f # #(X) for every limit ordinal # The first limit ....

P. Cousot and R. Cousot. A constructive characterization of the lattices of all retracts, pre-closure, quasi-closure and closure operators on a complete lattice.<F4.877e+05> Portugaliæ<F5.303e+05> Mathematica, 38(2):185--198, 1979.

.... such that for all x 2 L: f(f(x) f(x) idempotent) An upper closure operator (uco) on L is a retraction ae such that 8x 2 L: x ae(x) extensive) a lower closure operator (lco) on L is a retraction ffi such that 8x 2 L: ffi (x) x (reductive) More on closure operators can be found in [8]. Let h L; i and h L 0 ; 0 ; 0 ; 0 ; 0 ; 0 i be complete lattices. An upper Galois connection between L and L 0 is a pair of functions (ff; fl) such that 1. ff : L L 0 and fl : L 0 L 2. 8x 2 L : 8y 2 L 0 : ff(x) 0 y , x fl(y) An upper Galois ....

P. Cousot and R. Cousot. A constructive characterization of the lattices of all retracts, pre-closure, quasi-closure and closure operators on a complete lattice. Portugaliae Mathematica, 38(2):185--198, 1979.

....(x) x; if ff is a successor, then f ff (x) f(f ff Gamma1 (x) if ff is a limit ordinal, then f ff (x) fl ff f fl (x) The following lemma, due to Patrick and Radhia Cousot [3, Theorem 4.3] gives a constructive characterization for the lub t of closure operators. Lemma 4. 1 ([3]) Let fae i g i2I uco(C) Then, there exists ffi 2 Ord such that for any x 2 C, t i2I ae i ) x) fi ffi (y: i2I ae i (y) fi (x) The next lemma demonstrates that the lub of a nonempty family of closures fae i g i2I uco(C) is co additive iff the following condition holds: 8X C: t ....

P. Cousot and R. Cousot.A constructive characterization of the lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Portugal. Math., 38(2):185-- 198, 1979.

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P. Cousot and R. Cousot.A constructive characterization of the lattices of all retractions, preclosure, quasi-closure and closure operators on a complete lattice. Portugal. Math., 38(2):185-- 198, 1979.

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