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Ordinals[Power]

ordinal exponentiation

Ordinals[^]

ordinal exponentiation

&^

inert ordinal exponentiation

 Calling Sequence Power(a, b, ...) a ^ b a &^ b

Parameters

 a, b, ... - ordinals, nonnegative integers, or polynomials with positive integer coefficients

Description

 • The Power and ^ calling sequences perform exponentiation of the given ordinal numbers according to the rules of ordinal arithmetic. Let $a={\mathbf{\omega }}^{e}\cdot c+r$, where $c$ is a positive integer and $r=0$ or $e\succ \mathrm{degree}\left(r\right)$ in the strict ordering $\succ$ of ordinals.
 – ${0}^{b}=0$ if $b\ne 0$.
 – ${a}^{0}={1}^{b}=1$.
 – If $b$ is a positive integer, then ${a}^{b}=\underset{b\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}\mathrm{times}}{\underset{⏟}{a\cdot a\cdot \dots \cdot a}}$.
 – If $a=c\ge 2$ is a positive integer and $b=\mathbf{\omega }\cdot l+d$, where $d$ is a nonnegative integer, then ${a}^{b}={\mathbf{\omega }}^{l}\cdot {c}^{d}$.
 – If $e\ne 0$ and $\mathrm{tdegree}\left(b\right)\ne 0$, then ${a}^{b}={\mathbf{\omega }}^{e\cdot b}$.
 – If $b={b}_{1}+{b}_{2}$, then ${a}^{b}={a}^{{b}_{1}}\cdot {a}^{{b}_{2}}$.
 – If more than two arguments are specified, the powering will be performed right-associatively, that is, $\mathrm{Power}\left(a,b,c,\mathrm{...}\right)=\mathrm{Power}\left(a,\mathrm{Power}\left(b,c,\mathrm{...}\right)\right)$.
 • Mathematically, exponentiation of two ordinals ${a}^{b}$ corresponds to the set of all functions $f:b\to a$, between the corresponding well-orderings $b$ and $a$, with finite support, such that $f\left(x\right)\ne 0$ for only finitely many $x\in b$, together with the ordering defined by:

$f\prec g\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}⇔\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}\exists x\in b:\phantom{\rule[-0.0ex]{1.5ex}{0.0ex}}f\left(x\right)\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}{\prec }_{b}\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}g\left(x\right)\phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}\wedge \phantom{\rule[-0.0ex]{1.0ex}{0.0ex}}\forall y\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}{\succ }_{a}\phantom{\rule[-0.0ex]{0.5ex}{0.0ex}}x:\phantom{\rule[-0.0ex]{1.5ex}{0.0ex}}f\left(y\right)=g\left(y\right)$

 • In the two-argument case, if $a,b$ are both nonzero,  $a\ne 1$ and at least one of them is an ordinal data structure, that is, an ordinal number greater or equal to $\mathbf{\omega }$, then the result is an ordinal data structure. Otherwise, the result is a nonnegative integer or a polynomial with positive integer coefficients.
 • The &^ calling sequence is the inert form of ordinal exponentiation. No actual exponentiation is performed, but the result will be rendered as an inert power, with parentheses around the first argument.
 • Applying the value command will turn the inactive &. operator into the active . operator, causing the ordinal multiplication to be computed as described above.
 • The first argument $a$ can be a parametric ordinal. If it cannot be determined whether its leading or trailing coefficient is nonzero, an error may be raised.

Examples

 > $\mathrm{with}\left(\mathrm{Ordinals}\right)$
 $\left[{\mathrm{+}}{,}{\mathrm{.}}{,}{\mathrm{<}}{,}{\mathrm{<=}}{,}{\mathrm{Add}}{,}{\mathrm{Base}}{,}{\mathrm{Dec}}{,}{\mathrm{Decompose}}{,}{\mathrm{Div}}{,}{\mathrm{Eval}}{,}{\mathrm{Factor}}{,}{\mathrm{Gcd}}{,}{\mathrm{Lcm}}{,}{\mathrm{LessThan}}{,}{\mathrm{Log}}{,}{\mathrm{Max}}{,}{\mathrm{Min}}{,}{\mathrm{Mult}}{,}{\mathrm{Ordinal}}{,}{\mathrm{Power}}{,}{\mathrm{Split}}{,}{\mathrm{Sub}}{,}{\mathrm{^}}{,}{\mathrm{degree}}{,}{\mathrm{lcoeff}}{,}{\mathrm{log}}{,}{\mathrm{lterm}}{,}{\mathrm{\omega }}{,}{\mathrm{quo}}{,}{\mathrm{rem}}{,}{\mathrm{tcoeff}}{,}{\mathrm{tdegree}}{,}{\mathrm{tterm}}\right]$ (1)
 > $\mathrm{Power}\left(\right),\mathrm{Power}\left(0\right),\mathrm{Power}\left(0,0\right),\mathrm{Power}\left(0,0,0\right),\mathrm{Power}\left(0,0,0,0\right)$
 ${1}{,}{0}{,}{1}{,}{0}{,}{1}$ (2)
 > $\mathrm{Power}\left(\right),\mathrm{Power}\left(2\right),\mathrm{Power}\left(2,2\right),\mathrm{Power}\left(2,2,2\right),\mathrm{Power}\left(2,2,2,2\right)$
 ${1}{,}{2}{,}{4}{,}{16}{,}{65536}$ (3)
 > $\mathrm{length}\left(\mathrm{Power}\left(2,2,2,2,2\right)\right)$
 ${19729}$ (4)
 > $a≔\mathrm{\omega }·2+3$
 ${a}{≔}{\mathbf{\omega }}{\cdot }{2}{+}{3}$ (5)
 > ${a}^{0},a,{a}^{2},{a}^{3},{a}^{4}$
 ${1}{,}{\mathbf{\omega }}{\cdot }{2}{+}{3}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{2}{+}{\mathbf{\omega }}{\cdot }{6}{+}{3}{,}{{\mathbf{\omega }}}^{{3}}{\cdot }{2}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{6}{+}{\mathbf{\omega }}{\cdot }{6}{+}{3}{,}{{\mathbf{\omega }}}^{{4}}{\cdot }{2}{+}{{\mathbf{\omega }}}^{{3}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{6}{+}{\mathbf{\omega }}{\cdot }{6}{+}{3}$ (6)
 > ${0}^{\mathrm{\omega }},{1}^{\mathrm{\omega }},{2}^{\mathrm{\omega }},{3}^{\mathrm{\omega }},{4}^{\mathrm{\omega }}$
 ${0}{,}{1}{,}{\mathbf{\omega }}{,}{\mathbf{\omega }}{,}{\mathbf{\omega }}$ (7)
 > ${2}^{\mathrm{\omega }·2},{2}^{\mathrm{\omega }·2+3},{2}^{{\mathrm{\omega }}^{2}+3},{2}^{{\mathrm{\omega }}^{\mathrm{\omega }}+3}$
 ${{\mathbf{\omega }}}^{{2}}{,}{{\mathbf{\omega }}}^{{2}}{\cdot }{8}{,}{{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{8}{,}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{\mathbf{\omega }}}}{\cdot }{8}$ (8)
 > ${a}^{a}={\mathrm{\omega }}^{\mathrm{\omega }·2}·{a}^{3}$
 ${{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{3}}{\cdot }{2}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{2}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{1}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}}{\cdot }{3}{=}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{3}}{\cdot }{2}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{2}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{1}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}}{\cdot }{3}$ (9)

The inert exponentiation operator is useful for display purposes:

 > $\mathrm{result}≔a\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}&ˆ\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}a:$
 > $\mathrm{result}=\mathrm{value}\left(\mathrm{result}\right)$
 ${\left({\mathbf{\omega }}{\cdot }{2}{+}{3}\right)}^{{\mathbf{\omega }}{\cdot }{2}{+}{3}}{=}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{3}}{\cdot }{2}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{2}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}{+}{1}}{\cdot }{6}{+}{{\mathbf{\omega }}}^{{\mathbf{\omega }}{\cdot }{2}}{\cdot }{3}$ (10)

Parametric examples:

 > $b≔\mathrm{Ordinal}\left(\left[\left[2,x+1\right],\left[1,y\right],\left[0,z+1\right]\right]\right)$
 ${b}{≔}{{\mathbf{\omega }}}^{{2}}{\cdot }\left({x}{+}{1}\right){+}{\mathbf{\omega }}{\cdot }{y}{+}\left({z}{+}{1}\right)$ (11)
 > ${b}^{4}$
 ${{\mathbf{\omega }}}^{{8}}{\cdot }\left({x}{+}{1}\right){+}{{\mathbf{\omega }}}^{{7}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{6}}{\cdot }\left({x}{}{z}{+}{x}{+}{z}{+}{1}\right){+}{{\mathbf{\omega }}}^{{5}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{4}}{\cdot }\left({x}{}{z}{+}{x}{+}{z}{+}{1}\right){+}{{\mathbf{\omega }}}^{{3}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }\left({x}{}{z}{+}{x}{+}{z}{+}{1}\right){+}{\mathbf{\omega }}{\cdot }{y}{+}\left({z}{+}{1}\right)$ (12)
 > $o≔\mathrm{Ordinal}\left(\left[\left[\mathrm{\omega },3\right],\left[2,y\right],\left[0,z\right]\right]\right)$
 ${o}{≔}{{\mathbf{\omega }}}^{{\mathbf{\omega }}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{2}}{\cdot }{y}{+}{z}$ (13)
 > ${o}^{\mathrm{\omega }}$
 ${{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}}$ (14)
 > ${o}^{\mathrm{\omega }+1}={o}^{\mathrm{\omega }}·o$
 ${{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{\mathbf{\omega }}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{2}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}}{\cdot }{z}{=}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{\mathbf{\omega }}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{2}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}}{\cdot }{z}$ (15)
 > ${o}^{\mathrm{\omega }+2}$
 > ${\left(o+1\right)}^{\mathrm{\omega }+2}$
 ${{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{\mathbf{\omega }}{\cdot }{2}}{\cdot }{3}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{\mathbf{\omega }}{+}{2}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{\mathbf{\omega }}}{\cdot }\left({3}{}{z}{+}{3}\right){+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}{+}{2}}{\cdot }{y}{+}{{\mathbf{\omega }}}^{{{\mathbf{\omega }}}^{{2}}}{\cdot }\left({z}{+}{1}\right)$ (16)

Compatibility

 • The Ordinals[Power], Ordinals[^] and &^ commands were introduced in Maple 2015.