nLab
relative complement

Relative complements

Relative complements

Idea

The relative complement of AA in BB is like the complement of AA, but relative to BB.

Definitions

For subsets

Given two subsets SS and TT of a given ambient set XX, the relative complement of SS in TT is the set

TS={a|aTaS}. T \setminus S = \{ a \;|\; a \in T \;\wedge\; a \notin S \} .

That is, TST \setminus S consists of those elements of TT that are not elements of SS. This is also called the difference of the sets; TST \setminus S may even be written TST - S. (Compare the symmetric difference.)

Note that the relative complement of SS in the ambient set XX is simply the complement S\setminus S of SS; sometimes this is called the absolute complement. Conversely, TST \setminus S is the intersection of TT and S\setminus S.

In material set theory

In material set theory, all sets are subsets of the proper class VV of all sets (or all small things, if we might have atoms). In this case, there is no absolute complement (except as a proper class), but we can still form the relative complement:

TS={aT|aS}. T \setminus S = \{ a \in T \;|\; a \notin S \} .

In a lattice

Given a lattice LL and two elements xx and yy of LL, a relative complement of xx relative to yy is an element yxy \setminus x such that:

  • x(yx)= x \wedge (y \setminus x) = \bot (where \wedge is the meet in the lattice and \bot is the bottom of the lattice) and
  • yx(yx) y \leq x \vee (y \setminus x) (where \vee is the join in the lattice).

In a poset that is not a lattice, the same definition applies, with the existence of the relevant meet, join, and bottom bring required for the complement to exist. In any case, complements are unique. In a proset, we may speak of a complement, or the complement up to equivalence.

The term Boolean ring is sometimes used for something like a Boolean algebra in which there may be no top element (and therefore no complements) but still relative complements; compare the notions of ring vs algebra at sigma-algebra. That is, a Boolean ring is a bounded-below distributive lattice in which all relative complements exist.

In classical mathematics, this definition includes the previous ones; the power set of XX is a Boolean algebra, while the class of all sets is (in the sense above) a large Boolean ring.

Relative pseudocomplements

Given a lattice LL and two elements xx and yy of LL, a relative pseudocomplement of xx relative to yy is an element yxy \setminus x such that:

  • x(yx)= x \wedge (y \setminus x) = \bot and
  • given any zz such that xz= x \wedge z = \bot , yzyx y \wedge z \leq y \setminus x .

The same issues apply about posets, uniqueness, and prosets. A bounded-below distributive lattice in which all relative pseudocomplements exist may be called a Heyting ring, although I don’t know if anybody does. The set-theoretic examples are now valid even in constructive mathematics (although constructivists don’t add ‘pseudo’ in that context).

If a relative complement exists, then every relative pseudocomplement is a relative complement (and vice versa), so the same notation may be used. (Of course, some authors may intend for the notation to imply existence of a relative complement, so some care is still needed.)

Last revised on December 15, 2018 at 13:53:09. See the history of this page for a list of all contributions to it.