Relatively complemented lattice

Relatively complemented lattice

In mathematics, a relatively complemented lattice is a lattice "L" in which for all "a", "b", "c" in "L" with "a" ≤ "b" ≤ "c" there is some "x" in "L" such that "x" ∨ "b" = "c" and "x" ∧ "b" = "a". An element "x" with this property is a complement of "b" relative to the interval ["a","c"] .

Two particular cases are frequently seen:

*If "A" and "B" are sets with

::Asubseteq B

:then the complement of "A" relative to "B" (the "interval" involved is from the empty set to "B") is

::Bsetminus A=left{,xin B : x otin A, ight}.

*If the lattice is a Boolean algebra, then the complement of "b" relative to the interval ["a", "c"] is "a" ∨ (~ "b") ∧ "c". (In general, the expression "x" ∨ "y" ∧ "z" is ambiguous in Boolean algebra. But the fact that "a" ≤ "b" "c" removes the ambiguity in this case.) In the usual interpretation of Boolean algebra as a model of propositional logic, if "a" is a sufficient condition for "b" and "c" is a necessary condition for "b", the complement of "b" relative to the interval ["a", "c"] is the unique (up to logical equivalence) proposition "d" such that

:*"a" is sufficient for "d" and "c" is necessary for "d", and:*"d" becomes equivalent to [not "b"] if one learns that "a" is false and "c" is true.


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