Complementation

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

[b1]	B. A. Davey, H. A. Priestley, Introduction to lattices and order, 2nd ed. Cambridge University Press (2002) ISBN 978-0-521-78451-1

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

[b2]	Tsit-Yuen Lam, Lectures on Modules and Rings, Graduate Texts in Mathematics 189, Springer (1999) ISBN 0-387-98428-3

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

[b3]	Paul R. Halmos, Finite Dimensional Vector Spaces, Van Nostrand (1958)

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, 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

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