complement

The **complement** $\tilde S$ of a subset $S$ of a set $X$ is the set of ell elements of $X$ not contained in $S$:

$\tilde{S} = \{ a: X \;|\; a \notin S \}
.$

(Besides $\tilde{S}$, there are many other notations, such as $X - S$, $X \setminus S$, $\bar{S}$, $S^c$, $\neg{S}$, and so forth.)

Notice that $S \cap \tilde{S} = \empty$, while $S \cup \tilde{S} = X$ by the principle of excluded middle.

A **complement** of an element $S$ of a lattice is an element $T$ such that $S \wedge T = \bot$ and $S \vee T = \top$. Note that in general, complements need not be unique; for example, in the lattice of vector subspaces of a 2-dimensional vector space over a field $k$, a 1-dimensional subspace will have as many complements as there are elements of $k$. However, in some cases complements will be unique, for example in a distributive lattice, in which case it is denoted $\tilde{S}$ (or $\neg S$, etc.).

If every element has a complement, one speaks of a complemented lattice. Examples are Boolean algebras, and in fact complemented distributive lattices are the same thing as Boolean algebras (in the sense that the category of Boolean algebras is equivalent to the category of complemented distributive lattices).

More generally, the **pseudocomplement** of an element $S$ of a Heyting algebra is given by $\tilde{S} = S \Rightarrow \bot$. This satisfies $S \wedge \tilde{S} = \bot$ but not $S \vee \tilde{S} = \top$ in general. This case includes the complement of a subset even in constructive mathematics.

In another direction, the **complement** of a complemented subobject $S$ of an object $X$ in a coherent category is the unique subobject $\tilde{S}$ such that $S \cap \tilde{S}$ is the initial object and $S \cup \tilde{S} = X$.

The complement of a truth value (seen as a subset of the point) is called its *negation*.

Revised on July 15, 2017 02:48:23
by Mike Shulman
(76.167.222.204)