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

$A\subseteq B$

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

$B\setminus A=\left\{\,x\in B : x\not\in A\,\right\}.$

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