P. Dwinger. On the closure operators of a complete lattice. Indagationes Math., 16:560--563, 1954.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... in a complete lattice, relative pseudo complementation is equivalent to complete meet distributivity ( 1] and complete meet distributivity in uco(L) is equivalent to distributivity ( 6] Now, for uco(L) to be distributive, it is necessary and sufficient that L is a complete well ordered chain ([4]) This condition is also necessary and sufficient for uco(L) to be complemented ( 7] 2 Weak relative pseudo complement Before defining our notion of weak relative pseudo complement, we first recall some basic notions and terminology. We refer to [1, 5] for all the notions recalled in the ....

P. Dwinger. On the closure operators of a complete lattice. Indagationes Math., 16:560--563, 1954.

....) 3] In a complete lattice L, if x exists then x = fy 2 L j x y = g. If every x 2 L has the pseudo complement, L is pseudo complemented . It is worth noting that pseudo complementation is the only possible form of complementation for abstract interpretation. Indeed, it is well known [11, 20] that uco(C) is complemented (in the standard sense) i C is a complete well ordered chain, and this is a far too restrictive hypothesis for semantic domains. The following results [13, 12] provide sucient conditions on C such that uco(C) is pseudo complemented. Recall that a complete lattice C is ....

P. Dwinger. On the closure operators of a complete lattice. Indagat. Math., 16:560{ 563, 1954.

....) 4] In a complete lattice L, if x exists then x = fy 2 L j x y = g. If every x 2 L has the pseudo complement, L is pseudo complemented . It is worth noting that pseudo complementation is the only possible form of complementation for abstract interpretation. Indeed, it is well known [15, 26] that uco(C) is complemented (in the standard sense) i C is a complete well ordered chain, and this is a far too restrictive hypothesis for semantic domains. The following results [19, 17] provide sucient conditions on C such that uco(C) is pseudo complemented. Recall that a complete lattice C is ....

P. Dwinger. On the closure operators of a complete lattice. Indagat. Math., 16:560-563, 1954.

.... in a complete lattice, relative pseudo complementation is equivalent to complete meet distributivity ( 1] and complete meet distributivity in uco(L) is equivalent to distributivity ( 6] Now, for uco(L) to be distributive, it is necessary and sufficient that L is a complete well ordered chain ([4]) This condition is also necessary and sufficient for uco(L) to be complemented ( 7] 2 Weak relative pseudo complement Before defining our notion of weak relative pseudo complement, we first recall some basic notions and terminology. We refer to [1, 5] for all the notions recalled in the ....

P. Dwinger. On the closure operators of a complete lattice. Indagationes Math., 16:560--563, 1954.

.... for a complete lattice, relative pseudo complementation is equivalent to complete inf distributivity ( 1] and complete infdistributivity in uco(L) is equivalent to distributivity ( 7] Now, for uco(L) to be distributive, it is necessary (and sufficient) that L is a complete well ordered chain ([4]) This condition is also necessary and sufficient for uco(L) to be complemented ( 8] The plan of the paper is as follows: In x 2 we recall some basic notions and properties about (relative) pseudo complementation and closure operators. In x 3 we define the notion of weak relative ....

P. Dwinger. On the closure operators of a complete lattice. Indagationes Math., 16:560--563, 1954.

.... a complete lattice, relative pseudo complementation is equivalent to complete inf distributivity ( 1] and complete inf distributivity in uco(L) is equivalent to distributivity ( 7] Now, for uco(L) to be distributive, it is necessary (and sufficient) that L is a complete well ordered chain ([4]) This condition is also necessary and sufficient for uco(L) to be complemented ( 8] The plan of the paper is as follows: In x 2 we recall some basic notions and properties about (relative) pseudo complementation and closure operators. In x 3 we define the notion of weak relative ....

P. Dwinger. On the closure operators of a complete lattice. Indagationes Math., 16:560--563, 1954.

....product is additive if and only if uco(C ) is completely meet distributive, i.e. a complete Heyting algebra. This latter condition is equivalent to the weaker (finite) distributivity of uco(C ) cf. Morgado 1962] and this holds if and only if the concrete domain C is a complete chain (cf. [Dwinger 1954]) Obviously, being a complete chain for the concrete domain is not a reasonable hypothesis in semantics and program analysis. On the other hand, the general nonadditivity of the disjunctive completion refinement follows from Example 7.5 and Proposition 7.6. 7.4 Reordering Abstract Domains We ....

Dwinger, P. 1954. On the closure operators of a complete lattice. Indagat. Math. 16, 560--563.

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