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Let $( E , C )$ be a partially ordered vector space, [[#References|[a5]]], i.e. $E$ is a real [[Vector space|vector space]] with a convex cone $C$ defining the [[Partial order|partial order]] by $x \succ y$ if and only if $x - y \in C$. For $x \prec y$, the corresponding interval is $[ x , y ] = \{ u \in E : x \prec u \prec y \}$.

The (partially) ordered vector space $( E , C )$ has the Riesz decomposition property if $[ 0 , u ] + [ 0 , v ] = [ 0 , u + v ]$ for all $u , v \in C$, or, equivalently, if $[ x _ { 1 } , y _ { 1 } ] + [ x _ { 2 } , y _ { 2 } ] = [ x _ { 1 } + x _ { 2 } , y _ { 1 } + y _ { 2 } ]$ for all $x _ { 1 } \prec y _ { 1 }$, $x _ { 2 } \prec y _ { 2 }$.

A [[Riesz space|Riesz space]] (or [[Vector lattice|vector lattice]]) automatically has the Riesz decomposition property.

Terminology on this concept varies a bit: in [[#References|[a2]]] the property is referred to as the dominated decomposition property, while in [[#References|[a3]]] it is called the decomposition property of F. Riesz.

The Riesz decomposition property and the [[Riesz decomposition theorem|Riesz decomposition theorem]] are quite different (although there is a link) and, in fact, the property does turn up as an axiom used in axiomatic potential theory (see also [[Potential theory, abstract|Potential theory, abstract]]), see [[#References|[a1]]], where it is called the axiom of natural decomposition.

There is a natural non-commutative generalization to the setting of $C ^ { * }$-algebras, as follows, [[#References|[a4]]]. Let $x$, $y$, $z$ be elements of a [[C*-algebra|$C ^ { * }$-algebra]] $A$. If $x ^ { * } x \leq y y ^ { * } + z z ^ { * }$, then there are $u , v \in A$ such that $u ^ { * } u \leq y ^ { * } y$, $v ^ { * } v \leq x ^ { * } x$ and $x x ^ { * } = u u ^ { * } + v v ^ { * }$.

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  C. Constantinescu,   A. Cornea,   "Potential theory on harmonic spaces" , Springer  (1972)  pp. 104</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  W.A.J. Luxemburg,   A.C. Zaanen,   "Riesz spaces" , '''I''' , North-Holland  (1971)  pp. 73</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  P. Meyer-Nieberg,   "Banach lattices" , Springer  (1971)  pp. 3, Thm. 1.1.1</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  G.K. Pedersen,   "$C ^ { * }$-algebras and their automorphism groups" , Acad. Press  (1979)  pp. 14</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Y.-Ch. Wong,   K.-F. Ng,   "Partially ordered topological vector spaces" , Oxford Univ. Press  (1973)  pp. 9</td></tr></table>