exponentially decompose an ordinal number
ordinal or non-negative integer
(optional) literal keyword; either list (default) or inert
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 a1a2⋰an of a as an iterated power of ordinals and non-negative integers a1,a2,…,an 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⇔n=0, and if a=0, then n=1 and a1=0.
If a≥2 is an integer, then ak are all integers ≥2.
If ak≥2 is an integer, then it is not a perfect power, that is, it cannot be written as bc for integers b,c≥2.
If ak is not an integer, then either k=n=1 and a=ak=ω, or ak has at least two nonzero terms in the Cantor normal form.
If a is not an integer, then there is an index i∈1,…,n such that ai is not an integer and ai+1,…,an are all integers ≥2.
If i≥2, then a1=⋯=ai−2=2 and degree⁡ai≥1>0=tdegree⁡ai. (Moreover, either ai−1≥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 value⁡Decompose⁡a,output=inert=a.
The ordinal a can be parametric. However, if the complete decomposition cannot be computed in such a case, an error will be raised.
Using output=inert. The result can be verified using value.
Any ordinal ≠ω with a single term can be decomposed.
The following equality is not a decomposition into strictly smaller ordinals, and therefore ω is indecomposable.
2 &^ ω=2ω
More than one term.
b ≔ ω2+ω
a ≔ ωb+2+ωb+1
c ≔ a+`.`⁡ωb,3
p ≔ ω+22
q ≔ ω+3p
r ≔ ω+5q
f ≔ Ordinal⁡8,1,7,2,6,3,5,2,4,3,3,2,2,3,1,2,0,3
Non-negative integers can be decomposed as well.
u ≔ Ordinal⁡2,x,1,3⁢x,0,3
Error, (in Ordinals:-Decompose) cannot determine if x is nonzero
v ≔ Ordinal⁡ω+3,1,ω+2,x,ω+1,1
The Ordinals[Decompose] command was introduced in Maple 2015.
For more information on Maple 2015 changes, see Updates in Maple 2015.
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