nLab relative complement

Idea

The relative complement of $A$ in $B$ is like the complement of $A$, but relative to $B$.

Definitions

For subsets

Given two subsets $S$ and $T$ of a given ambient set $X$, the relative complement of $S$ in $T$ is the set

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

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

Note that the relative complement of $S$ in the ambient set $X$ is simply the complement $\setminus S$ of $S$; sometimes this is called the absolute complement. Conversely, $T \setminus S$ is the intersection of $T$ and $\setminus S$.

In constructive mathematics, the above definitions still holds for any two decidable subsets of a given ambient set $X$.

In material set theory

In material set theory, all sets are subsets of the proper class $V$ 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:

$$T \setminus S = \{ a \in T \;|\; a \notin S \} .$$

In a lattice

Given a lattice $L$ and two elements $x$ and $y$ of $L$, a relative complement of $x$ relative to $y$ is an element $y \setminus x$ such that:

• $x \wedge (y \setminus x) = \bot$ (where $\wedge$ is the meet in the lattice and $\bot$ is the bottom of the lattice) and
• $y \leq x \vee (y \setminus x)$ (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 $X$ is a Boolean algebra, while the class of all sets is (in the sense above) a large Boolean ring.

Relative pseudocomplements

Given a lattice $L$ and two elements $x$ and $y$ of $L$, a relative pseudocomplement of $x$ relative to $y$ is an element $y \setminus x$ such that:

• $x \wedge (y \setminus x) = \bot$ and
• given any $z$ such that $x \wedge z = \bot$, $y \wedge z \leq y \setminus x$.

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.)