Relatively complemented lattice
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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
- <math>A\subseteq B</math>
- then the complement of A relative to B (the interval involved is from the empty set to B) is
- <math>B\setminus A=\left\{\,x\in B : x\not\in A\,\right\}.</math>
- 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.