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

$Bsetminus\; A=left\{,xin\; B\; :\; xnotin\; A,right\}.$

• 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.