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A couple of examples
Jul. 18th, 2006 08:44 amConsider a set of all subsets of some set X, P(X), partially ordered by inclusion, ⊆. Consider a function "conjunction with U" from P(X) to P(X), mapping each set V to U ∩ V. The upper adjoint of this function will be a function mapping each set W to the largest set V, such that U ∩ V ⊆ W. This largest set V is (X \ U) ∪ W, which is why the upper adjoint is called "the implication from U".
A one element set X yields a familiar Boolean implication, if its subset X encodes true, and empty subset, ∅, encodes false, yielding partial order false ≤ true.
As another example, consider the set R+ of non-negative reals with added +∞, but with the order being the reverse of the usual: +∞ ≤' x ≤' 0. Consider a function "addition to x" mapping R+ to R+ as follows: y maps to x + y. The upper adjoint of this function will be a function mapping each z to the largest (in our reversed order, the largest is closest to zero) number y, such that x + y ≤' z.
This "largest" (closest to 0) number is z - x, when z ≤' x, and 0 otherwise. We can write this operation as mapping z to z -' x. When people want to view all this logically, they, sometimes, still call this operation "an implication from x", thinking about + as a generalized conjunction.
A one element set X yields a familiar Boolean implication, if its subset X encodes true, and empty subset, ∅, encodes false, yielding partial order false ≤ true.
As another example, consider the set R+ of non-negative reals with added +∞, but with the order being the reverse of the usual: +∞ ≤' x ≤' 0. Consider a function "addition to x" mapping R+ to R+ as follows: y maps to x + y. The upper adjoint of this function will be a function mapping each z to the largest (in our reversed order, the largest is closest to zero) number y, such that x + y ≤' z.
This "largest" (closest to 0) number is z - x, when z ≤' x, and 0 otherwise. We can write this operation as mapping z to z -' x. When people want to view all this logically, they, sometimes, still call this operation "an implication from x", thinking about + as a generalized conjunction.
