If output=list (the default), a list of ordinals and non-negative integers is returned. Unless a=0 or a=1, any integers in the list are strictly greater than $1$.

Otherwise, if output=inert is specified, an inert exponentiation of ordinal numbers using the inert operator &^ is returned.

The Decompose(a) calling sequence computes an exponential normal form $a_1^{a_2^{{}^{{⋰}^{a_n}}}}$ of $a$ as an iterated power of ordinals and non-negative integers $a_1\,a_2\,\dots \,a_n$ that cannot be decomposed any further as a power of strictly smaller ordinals.

The composition factors have the following additional properties, which ensure uniqueness of the decomposition.

Trivial cases: $a\=1\iff n\=0$, and if $a=0$, then $n=1$ and $a_1=0$.

If $a\ge 2$ is an integer, then $a_k$ are all integers $\ge 2$.

If $a_k\ge 2$ is an integer, then it is not a perfect power, that is, it cannot be written as $b^c$ for integers $b\,c\ge 2$.

If $a_k$ is not an integer, then either $k\=n\=1$ and $a\=a_k\=\mathbf{\omega }$, or $a_k$ has at least two nonzero terms in the Cantor normal form.

If $a$ is not an integer, then there is an index $i\in \left\{1\,\dots \,n\right\}$ such that $a_i$ is not an integer and $a_{i\+1}\,\dots \,a_n$  are all integers $\ge 2$.

If $i\ge 2$, then $a_1\=\cdots \=a_{i-2}\=2$ and $\mathrm{degree}\left(a_i\right)\ge 1\>0\=\mathrm{tdegree}\left(a_i\right)$. (Moreover, either $a_{i-1}\ge 2$ is an integer, or it has at least two nonzero terms.)

Exponential decomposition is a one-sided inverse of powering, in the sense that $\mathrm{value}\left(\mathrm{Decompose}\left(a\,\mathrm{output}\=\mathrm{inert}\right)\right)\=a$.

The ordinal $a$ can be parametric. However, if the complete decomposition cannot be computed in such a case, an error will be raised.

$\mathrm{with}\left(\mathrm{Ordinals}\right)$

$\left[\mathrm{`+`}\&comma;\mathrm{`.`}\&comma;\mathrm{`<`}\&comma;\mathrm{<=}\&comma;\mathrm{Add}\&comma;\mathrm{Base}\&comma;\mathrm{Dec}\&comma;\mathrm{Decompose}\&comma;\mathrm{Div}\&comma;\mathrm{Eval}\&comma;\mathrm{Factor}\&comma;\mathrm{Gcd}\&comma;\mathrm{Lcm}\&comma;\mathrm{LessThan}\&comma;\mathrm{Log}\&comma;\mathrm{Max}\&comma;\mathrm{Min}\&comma;\mathrm{Mult}\&comma;\mathrm{Ordinal}\&comma;\mathrm{Power}\&comma;\mathrm{Split}\&comma;\mathrm{Sub}\&comma;\mathrm{`^`}\&comma;\mathrm{degree}\&comma;\mathrm{lcoeff}\&comma;\log \&comma;\mathrm{lterm}\&comma;\mathrm{\omega }\&comma;\mathrm{quo}\&comma;\mathrm{rem}\&comma;\mathrm{tcoeff}\&comma;\mathrm{tdegree}\&comma;\mathrm{tterm}\right]$

$\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }}\right)$

$\left[2\&comma;2\&comma;2\&comma;\mathbf{\omega }\&plus;1\right]$

$\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }}\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(2\right)}^{{\left(2\right)}^{{\left(2\right)}^{\mathbf{\omega }\&plus;1}}}$

$\mathrm{value}\left(\right)$

${\mathbf{\omega }}^{\mathbf{\omega }}$

$\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }+1}\right)$

$\left[2\&comma;{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\right]$

$\mathrm{Decompose}\left(\mathrm{\omega }·3\right)$

$\left[3\&comma;\mathbf{\omega }\&plus;1\right]$

$\mathrm{Decompose}\left({\mathrm{\omega }}^3\right)$

$\left[2\&comma;3\&comma;\mathbf{\omega }\&plus;1\right]$

$\mathrm{Decompose}\left({\mathrm{\omega }}^2·3\right)$

$\left[3\&comma;\mathbf{\omega }\cdot 2\&plus;1\right]$

$\mathrm{Decompose}\left({\mathrm{\omega }}^{\mathrm{\omega }+1}·2\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(2\right)}^{{\left(\mathbf{\omega }\&plus;1\right)}^2}$

$\mathrm{value}\left(\right)$

${\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}\cdot 2$

$2\&ˆ\mathrm{\omega }=2^{\mathrm{\omega }}$

${\left(2\right)}^{\mathbf{\omega }}=\mathbf{\omega }$

$\mathrm{Decompose}\left(\mathrm{\omega }\right)$

$\left[\mathbf{\omega }\right]$

$b≔{\mathrm{\omega }}^2+\mathrm{\omega }$

$b≔{\mathbf{\omega }}^2\&plus;\mathbf{\omega }$

$a≔{\mathrm{\omega }}^{b+2}+{\mathrm{\omega }}^{b+1}$

$a≔{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;2}\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;1}$

$\mathrm{Decompose}\left(a\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^2\&plus;\mathbf{\omega }\right)}^{{\left(\mathbf{\omega }\&plus;1\right)}^2}$

$c≔a+{\mathrm{\omega }}^b·3$

$c≔{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;2}\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;1}\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }}\cdot 3$

$\mathrm{Decompose}\left(c\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^2\&plus;\mathbf{\omega }\&plus;3\right)}^{{\left(\mathbf{\omega }\&plus;1\right)}^2}$

$p≔{\left(\mathrm{\omega }+2\right)}^2$

$p≔{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;2$

$\mathrm{Decompose}\left(p\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(\mathbf{\omega }\&plus;2\right)}^2$

$q≔{\left(\mathrm{\omega }+3\right)}^p$

$q≔{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;2}\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;1}\cdot 3\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2}\cdot 3$

$\mathrm{Decompose}\left(q\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(\mathbf{\omega }\&plus;3\right)}^{{\left(\mathbf{\omega }\&plus;2\right)}^2}$

$r≔{\left(\mathrm{\omega }+5\right)}^q$

$r≔{\mathbf{\omega }}^{{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;2}\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;1}\cdot 3\&plus;{\mathbf{\omega }}^{{\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2}\cdot 3}$

$\mathrm{Decompose}\left(r\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(2\right)}^{{\left(\mathbf{\omega }\&plus;3\right)}^{{\left(\mathbf{\omega }\&plus;2\right)}^2}}$

$f≔\mathrm{Ordinal}\left(\left[\left[8\&comma;1\right]\&comma;\left[7\&comma;2\right]\&comma;\left[6\&comma;3\right]\&comma;\left[5\&comma;2\right]\&comma;\left[4\&comma;3\right]\&comma;\left[3\&comma;2\right]\&comma;\left[2\&comma;3\right]\&comma;\left[1\&comma;2\right]\&comma;\left[0\&comma;3\right]\right]\right)$

$f≔{\mathbf{\omega }}^8\&plus;{\mathbf{\omega }}^7\cdot 2\&plus;{\mathbf{\omega }}^6\cdot 3\&plus;{\mathbf{\omega }}^5\cdot 2\&plus;{\mathbf{\omega }}^4\cdot 3\&plus;{\mathbf{\omega }}^3\cdot 2\&plus;{\mathbf{\omega }}^2\cdot 3\&plus;\mathbf{\omega }\cdot 2\&plus;3$

$\mathrm{Decompose}\left(f\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^2\&plus;\mathbf{\omega }\cdot 2\&plus;3\right)}^{{\left(2\right)}^2}$

$\mathrm{Decompose}\left(2417851639229258349412352\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(2\right)}^{{\left(3\right)}^{{\left(2\right)}^2}}$

$u≔\mathrm{Ordinal}\left(\left[\left[2\&comma;x\right]\&comma;\left[1\&comma;3x\right]\&comma;\left[0\&comma;3\right]\right]\right)$

$u≔{\mathbf{\omega }}^2\cdot x\&plus;\mathbf{\omega }\cdot \left(3x\right)\&plus;3$

$\mathrm{Decompose}\left(u\right)$

Error, (in Ordinals:-Decompose) cannot determine if x is nonzero

$\mathrm{Decompose}\left(\mathrm{Eval}\left(u\&comma;x=x+1\right)\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left(\mathbf{\omega }\cdot \left(x+1\right)\&plus;3\right)}^2$

$v≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega }+3\&comma;1\right]\&comma;\left[\mathrm{\omega }+2\&comma;x\right]\&comma;\left[\mathrm{\omega }+1\&comma;1\right]\right]\right)$

$v≔{\mathbf{\omega }}^{\mathbf{\omega }\&plus;3}\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;2}\cdot x\&plus;{\mathbf{\omega }}^{\mathbf{\omega }\&plus;1}$

$\mathrm{Decompose}\left(v\&comma;\mathrm{output}=\mathrm{inert}\right)$

${\left({\mathbf{\omega }}^3\&plus;{\mathbf{\omega }}^2\cdot x\&plus;\mathbf{\omega }\right)}^{\mathbf{\omega }\&plus;1}$

The Ordinals[Decompose] command was introduced in Maple 2015.

For more information on Maple 2015 changes, see Updates in Maple 2015.