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An operation which brings a subset $M$ of a given set $X$ into correspondence with another subset $N$ so that if $M$ and $N$ are known, it is possible in some way to reproduce $X$. Depending on the structure with which $X$ is endowed, one distinguishes various definitions of complementation, as well as various methods of reconstituting $X$ from $M$ and $N$.

===Sets===
In the general theory of sets the complement of a subset $M$ (or complementary subset, relative complement) in a set $X$ is the subset $\complement_X M$ (or $\complement M$ if $X$ is assumed, or $X \setminus M$) consisting of all elements $x \in X$ not belonging to $M$; an important property is the [[duality principle]] (one of the [[De Morgan laws]]):
$$
\complement \bigcup_{\xi} M_\xi = \bigcap_\xi \complement M_\xi
$$

===Lattices===
Let $L$ be a [[lattice]] with 0 and 1and $n$ an element of $L$   Then $m$ is a complement of $n$ if $m \vee n = 1$, $m \wedge n = 0$.  In a [[complemented lattice]] each element has at least one complement; a [[distributive lattice]] has the property that each element has at most one complement.  A [[Boolean lattice]] is a distributive lattice in which each element has a (unique) complement.  

====References====
<table>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  B. A. Davey, H. A. Priestley, ''Introduction to lattices and order'', 2nd ed. Cambridge University Press  (2002) ISBN 978-0-521-78451-1</TD></TR>
</table>

===Modules===
Let $M$ be a [[module]] over a ring and $N$ a submodule.
An '''intersection complement''' of $N$ is a submodule $C$ such that $C \cap N = \{0\}$ and $C$ is maximal with respect to this condition.  
An '''addition complement''' of $N$ is a submodule $C$ such that $C + N = M$ and $C$ is minimal with respect to this condition.
A '''direct complement''' of $N$ is a submodule $C$ such that $M = C \oplus N$: that is, $C \cap N = \{0\}$ and $C + N = M$.

====References====
<table>
<TR><TD valign="top">[b2]</TD> <TD valign="top">  Tsit-Yuen Lam, ''Lectures on Modules and Rings'', Graduate Texts in Mathematics '''189''', Springer (1999) ISBN 0-387-98428-3</TD></TR>
</table>

===Linear spaces===
Let $X$ have a structure of a [[linear space]] and let $M$ be a subspace of $X$. A subspace $N \subset X$ is said to be a direct algebraic complement ('''direct''' or '''algebraic complement''', for short) of $M$ if any $x \in X$ can be uniquely represented as $x = y+z$, $y \in M$, $z \in N$. This is equivalent to the conditions $X = M + N$; $M \cap N = \{0\}$. Any subspace of $X$ has an algebraic complement, but this complement is not uniquely determined.

===Inner product spaces===
In an [[inner product space]] $V$, the '''orthogonal complement''' of a subspace $N$ consists of all vectors orthogonal to every element of $N$:
$$
N^\perp = \{ y \in V : \forall x\in N\ (y,x) = 0 \}
$$
where $(\,,\,)$ is the inner product on $V$.

If $V$ is finite dimensional then $V$ is an orthogonal [[direct sum]], $V = N \oplus N^\perp$ and $(N^\perp)^\perp = N$.  

====References====
<table>
<TR><TD valign="top">[b3]</TD> <TD valign="top">  Paul R. Halmos, ''Finite Dimensional Vector Spaces'', Van Nostrand (1958)</TD></TR>
</table>

===Linear topological spaces===
Let $(X,\tau)$ be a [[linear topological space]] and let $X$ be the direct algebraic sum $X = L + N$ of subspaces $L$ and $N$, regarded as linear topological spaces with the induced topology. The one-to-one mapping $(y,z) \mapsto x+y$ of the Cartesian product $L \times N$ onto $X$, which is continuous by virtue of the linearity of the topology $\tau$, does not have, in general, a continuous inverse. If this mapping is a homeomorphism, i.e. if $X$ is the direct topological sum of the spaces $L$ and $N$, the subspace $N$ is said to be the direct topological complement of the subspace $L$, the latter being known as a ''complementable'' subspace. Not all subspaces in an arbitrary linear topological space, not even the finite-dimensional ones, are complementable. The following necessary and sufficient condition for complementability holds: The subspace $L$ is topologically isomorphic to $X/N$, where $N$ is an algebraic complement of $L$. This criterion entails the following sufficient conditions for complementability: $L$ is closed and has finite codimension; $X$ is locally convex and $L$ is finite-dimensional; etc.

===Hilbert spaces===
A special case of topological complementation is the '''orthogonal complement''' of a subspace $M$ of a Hilbert space $H$. This is the set
$$
N^\perp = \{ x \in H : (x,y) = 0 \ \text{for all}\ y \in N \}
$$
which is a closed subspace of $H$. An important fact in the theory of Hilbert spaces is that any closed subspace of a Hilbert space has an orthogonal complement, $H = N \oplus N^\perp$.

===Vector lattices===
Finally, let $X$ be a conditionally order-complete vector lattice: a [[K-space|$K$-space]], a partially ordered real vector space with an order relation ([[vector lattice]]) that is a [[conditionally-complete lattice]], cf. [[Semi-ordered space]]. The totality of elements of the form
$$
M^d = \{ x \in X : |x| \wedge |y| = 0\ \text{for all}\ y\in M \}\,,
$$
which is a linear subspace of $X$, is said to be the '''disjoint complement''' of the set $M \subset X$. If $M$ is a linear subspace, then, in the general case, $X \neq M + M^d$, but if $M$ is a component (also known as a band or an order-complete ideal), i.e. a linear subspace such that $x \in M$ and $|y| \le |x|$ imply that $y \in M$, and such that $M$ is closed with respect to least upper and greatest lower bounds, then $X = M + M^d$.  The set $M^d$ is a component for any $M$; $M^{dd} = (M^d)^d$ is the smallest component containing the set $M$.

===References===
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Theory of sets" , Addison-Wesley  (1968)  (Translated from French)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Topological vector spaces" , Addison-Wesley  (1977)  (Translated from French)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  G. Birkhoff,   "Lattice theory" , ''Colloq. Publ.'' , '''25''' , Amer. Math. Soc.  (1973)</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  A.P. Robertson,   W.S. Robertson,   "Topological vector spaces" , Cambridge Univ. Press  (1964)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  B.Z. Vulikh,   "Introduction to the theory of partially ordered spaces" , Wolters-Noordhoff  (1967)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Köthe,   "Topological vector spaces" , '''1''' , Springer  (1969)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  W.A.J. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1971)</TD></TR>
</table>