PART 0 -- Somewhat Standard Terminology:
(0.1) Definition of: The n-fold iteration of f evaluated at (x). If f : X -> X, then f^(n)(x) = f(f^(n-1)(x)) and f^(1)(x) = f(x).
(0.2) Definition of: The n-fold argument-m iteration of g evaluated at (x_1, x_2, ..., x_N). If g : X^N -> X, then g^(n, m)(...) = g(..., x_(m-1), g^(n-1, m)(...), x_(m+1), ...) and g^(1, m)(...) = g(...).
(0.3) Definition of: A "power" function. f(x) = x^n.
(0.4) Definition of: An "exponential" function. f(x) = b^x.
(0.5) Definition of: A "super-power" function. f(x) = x^^n.
(0.6) Definition of: A "super-exponential" function. f(x) = b^^x.
PART 1 -- Questions about Hyper-operators:
(1.1) Term for: the sequence { x + y, xy, x^y, x^^y, ... }?
Let f_1(x, y) = x + y, then f_n : x,y -> f_n(x, y) is the
y-fold argument-2 iteration of f_(n-1) evaluated at (x, 1).
see C.A. Rubtsov and G.F. Romerio,
"Ackermann's Function and New Arithmetical Operations",
http://www.rotarysaluzzo.it/filePDF/Iperoperazioni%20(1).pdf .
see E.W. Weisstein, "CRC Concise Encyclopedia of Mathematics"
see http://mathworld.wolfram.com/ArrowNotation.html .
see http://mathworld.wolfram.com/DownArrowNotation.html .
What is this sequence to be called?
Possibilities:
- "Ackermann function (3-argument)"
- "Ackermann operator sequence"
- "Grzegorczyk hierarchy"
- "R-hyper-operator sequence" (R for right-associative)
- "hyper-n operator sequence"
- "hyper-operator sequence" (I recommend this)
- "hyper-operators"
- (other)
(1.2) Terms for: any member of (1.1)?
Generally, what should we call these operations and their inverses?
Possibilities:
- {R-hyper-operators, R-hyper-logarithms, R-hyper-roots}.
- {hyper-operators, hyper-logarithms, hyper-roots}.
- (other)
(1.3) Terms for: the n-th member of (1.1)?
Specifically what should we call the n-th member of (1.1) and its two inverses?
Possibilities:
- {R-hyper-n, R-hyper-n-logarithms, R-hyper-n-roots}
- {hyper-n, hyper-n-logarithms, hyper-n-roots} (I recommend this)
- {hyper-n, hyper-logarithms (rank n), hyper-roots (rank n)} (Rubstov)
- (other)
(1.4) Notation for: the n-th member of (1.1) and inverses?
Which one of these notations should be used?
Possibilities:
- Knuth's up-arrow notation, TeX: {x \uparrow^{n-2} y} for
hyper-operators, and Vardi's down-arrow notation,
TeX: {x \downarrow^{n-2} y} for hyper-logarithms,
this leaves no room for a notation for hyper-roots.
- Munafo's parenthesis notation, TeX: {x^{(n)} y} (no inverse notation)
- Rubstov's boxed notation, TeX: {x \boxed{n} y} with
Rubstov's notation for inverses (not expressible with ASCII)
- {hyper(x, n, y), hyperlog(x, n, z), hyperroot(z, n, y)}
- {Hy_n(x, y), Hylog_n(x, z), Hyrt_n(z, y)} (I recommend this)
- (other)
(1.5) Term for: the sequence { x + y, xy, x^y, x^(x^(y-1)), ... }?
Let f_1(x, y) = x + y, then f_n : x,y -> f_n(x, y) is the
(y-1)-fold argument-1 iteration of f_(n-1) evaluated at (x, x).
What is this sequence to be called?
Possibilities:
- "L-hyper-operator sequence" (L for left-associative)
- "lower-hyper-operator sequence" (I recommend this)
- "hypo-operator sequence"
- (other)
(1.6) Terms for: any member of (1.5)?
Generally, what should we call these operations and their inverses?
Possibilities:
- {lower-hyper-operators, lower-hyper-logarithms, lower-hyper-roots}.
- {hypo-operators, hypo-logarithms, hypo-roots}.
- (other)
(1.7) Terms for: the n-th member of (1.5)?
Specifically what should we call the n-th member of (1.5) and its two inverses?
Possibilities:
- {lower-hyper-n, lower-hyper-n-logarithms, lower-hyper-n-roots}
- {lower-n, lower-n-logarithms, lower-n-roots} (I recommend this)
- {hypo-n, hypo-n-logarithms, hypo-n-roots}
- (other)
(1.8) Notation for: the n-th member of (1.5)?
- Munafo's parenthesis notation, TeX: {x_{(n)} y}
- {hypo(x, n, y), hypolog(x, n, z), hyporoot(z, n, y)}
- {Lo_n(x, y), Lolog_n(x, z), Lort_n(z, y)}
- (other)
(1.9) Term for: The sequence { x*y + z, z*x^y, x^^y@z } ?
Let f_1(x, y, z) = x*y + z, then f_n : x,y,z -> f_n(x, y, z) is
the y-fold argument-2 iteration of f_(n-1) evaluated at (x, z).
If g(y) = x + y then g^y(z) = x*y + z.
If g(y) = x*y then g^y(z) = z*x^y.
If g(y) = x^y then g^y(z) = x^^y@z.
Possibilities:
- "auxiliary hyper-n operator sequence"
- "auxiliary hyper-operator sequence"
- "iterated hyper-n operator sequence"
- "iterated hyper-operator sequence"
- "iterated hyper-operators"
- (other)
(1.10) Term for: The n-th member of (1.9) ?
Possibilities:
- "auxiliary hyper-(n-1)"
- "iterated hyper-n"
- (other)
(1.11) Term for: Any member of (1.9) ?
Possibilities:
- "auxiliary hyper-operators"
- "iterated hyper-operators"
- (other)
COMMENT: There are also more general sets of hyper-like-operators. One is the tree of hyper-like-operators investigated by Mueller, which could be called mixed-hyper-operators of which (1.1) and (1.5) are special cases. Mueller's mixed-hyper-operators are defined by left-associative AND right-associative iteration, where after hyper-3, there are 2 hyper-4's (L-hyper-4, R-hyper-4), 4 hyper-5's (LL-hyper-5, LR-hyper-5, RL-hyper-5, RR-hyper-5), etc. Just as "R-hyper-4" is usually called "tetration", the operator "RR-hyper-5" is usually what people call "pentation". For a more general hyper-like-operator, Trappmann has investigated all possible groupings of non-associative operators and defined a "successor" to an operator as a product between a binary tree and a base, such that Mueller's mixed-hyper-operators are special cases of what could be called Trappmann's binary-tree-hyper-operators. Trappmann's operators when taken to a hyper-like-operator extreme would be defined as H : B^n * N -> N, where B is the set of binary trees representing association. Since this is still an area of preliminary research, I will not look for standard terminology yet.
see Mueller, http://www.math.tu-berlin.de/~mueller/reihenalgebra.pdf
see Trappmann, http://blafoo.de/tree-aoc/main1176.pdf
PART 2 -- Questions about Towers:
[heterogeneous hyper4]?
[heterogeneous hyper4 function]?
[tower]?
[subtower]?
[partial subtower]?
[infinite subtower]?
[n-ary tower notation]?
(2.1) Notation for: tetration ?
- {}^y x
- x ^^ y [Knuth up arrow]
- (other)
(2.2) Notation for: hyper-4-logarithms, super-logarithms ?
- slog_x(z) iff x^^y = z (Rubstov and Wikipedia) (I recommend this)
- (other)
(2.3) Notation for: hyper-4-roots, super-roots ?
see C.A. Rubtsov and G.F. Romerio,
"Ackermann's Function and New Arithmetical Operations",
http://www.rotarysaluzzo.it/filePDF/Iperoperazioni%20(1).pdf
- srt_y(z) iff x^^y = z (I recommend this)
- Rubstov's notation (not expressible in ASCII)
- (other)
(2.1) Notation for: iterated exponentials ?
- exp_b^n(x)
- (other)
(2.2) Notation for: auxiliary hyper-4-logarithms, auxiliary super-logarithms ?
- slog_x(z) - slog_x(a)
- (other)
(2.3) Notation for: hyper-4-roots, super-roots ?
see C.A. Rubtsov and G.F. Romerio,
"Ackermann's Function and New Arithmetical Operations",
http://www.rotarysaluzzo.it/filePDF/Iperoperazioni%20(1).pdf
- srt_y(z) iff x^^y = z (I recommend this)
- Rubstov's notation (not expressible in ASCII)
- (other)
PART 3 -- Questions about Continuous Iteration:
[Bell matrix]?
[Carleman matrix]?
[continuous iterate]?
PART 4 -- Terminology Questions:
8) Term for: ((z^a)^a)^...^a (with b a's)
- "left-associative iterated hyper-3"
- "lower-hyper-4"
- "lower tetration"
- "iterated powers" (I recommend this)
9) Term for: a^^b = a^(a^(a^...^a)) (with b a's)
- "hyper-4"
- "tetration" (I recommend this)
- "a hyperpower function" (MacDonnell)
- "iterated exponentials" (I discourage this)
- "exponential towers" (Galidakis)
- "power towers" (Weisstein)
- "super-exponentiation"
- "hyper-exponentiation"
- "generalized exponentials" (Walker)
- (other)
10) Term for: a^^inf = a^(a^(a^...))
- "infinitely iterated exponentials" (Knoebel)
- "infinite exponentials" (Galidakis) (I discourage this)
- "infinite tetration" (I recommend this)
- "the hyperpower function" (MacDonnell)
- (other)
11) Term for: f(z) = a^(a^(a^...^a^z))
- "iterated exponentials" (I recommend this)
- "iterated exponential function"
- "incomplete power tower"
- "auxiliary tetration"
- (other)
12) Term for: a1^(a2^(a3^(a4^...)))
see G. Bachman, "Convergence of Infinite Exponentials"
see D.L. Shell, "On the convergence of infinite exponentials"
see W.J. Thron, "Convergence of infinite exponentials with complex elements"
see D.F. Barrow, "Infinite exponentials"
- "infinitely nested exponentials"
- "infinite exponentials" (Barrow, Thron, Shell, Bachman) (I recommend this)
- "general infinite power towers"
- (other)
13) Term for: a1^(a2^(a3^...^an))
- "nested exponentials" (I recommend this)
- "general power towers"
- (other)
14) Term for: f(z) = a1^(a2^(a3^...^an^z))
see D.F. Barrow, "Infinite exponentials"
- "nested exponential function"
- "margerin exponentials" (I recommend this)
- "nth margerin function of z" (Barrow)
15) Term for: f(x, z) = y if and only if x^^y = z ?
see Peter Walker, "Infinitely Differentiable Generalized
Logarithmic and Exponential Functions"
see Tetraspace Forums,
"Tetration and other large number sequences",
http://tetraspace.alkaline.org/forum/viewtopic.php?t=248
see C.A. Rubtsov and G.F. Romerio,
"Ackermann's Function and New Arithmetical Operations",
http://www.rotarysaluzzo.it/filePDF/Iperoperazioni%20(1).pdf
- "super-logarithms" (Rubstov) (I recommend this)
- "tetra-logarithms" (Tetraspace) (I discourage this,
because it conflicts with poly-logarithms)
- "generalized logarithms" (Walker)
- (other)
16) Term for: f(z, y) = x if and only if x^^y = z ?
- "super-roots" (I recommend this)
- "tetra-roots"
- (other)
17) Term for: f(x, z, a) = y if and only if x^(x^(x^...^x^a)) = z (with y x's) ?
- "auxiliary super-logarithms"
- (other)
18) Term for: f(z, y, a) = x if and only if x^(x^(x^...^x^a)) = z (with y x's) ?
- "auxiliary super-roots"
- (other)
19) Term for: f(x, z, n) = y if and only if x ^[n-2] y = z (Knuth's up-arrow) ?
see http://mathworld.wolfram.com/DownArrowNotation.html
On MathWorld they use an odd notation for hyper-logarithms,
which conflicts with Mueller's mixed-arrow notation, and
prevents extension to hyper-roots as well.
- "hyper-n-logarithm" (I recommend this)
- "hyper-logarithms"
- (other)
20) Term for: f(z, y, n) = x if and only if x ^[n-2] y = z (Knuth's up-arrow) ?
- "hyper-n-root" (I recommend this)
- "hyper-roots"
- (other)
21) Term for: f(x, z, a, n) = y if f(y) = hyper-n(x, y) and f^y(a) = z ?
- "auxiliary hyper-logarithms"
- "(iterated hyper-n)-logarithms"
- (other)
22) Term for: f(z, y, a, n) = x if f(y) = hyper-n(x, y) and f^y(a) = z ?
- "auxiliary hyper-roots"
- "(iterated hyper-n)-roots"
- (other)
23) Term for: RESULT of iterated exponentials, nested exponentials, etc. ?
Rationale: the result of + is "sum", the result of * is "product".
If an n-ary operator is to be defined for exponentials, then
it should be named accordingly, because of "n-ary sums", etc.
- "exponentials"
- "exponential towers"
- "towers" (I recommend this)
- (other)
PART 5 -- Notation Questions:
(5.1) Notation for: nested exponentials, general power towers ?
see B.W. Brunson "The Partial Order of Iterated Exponentials"
see D.L. Shell, "On the convergence of infinite exponentials"
see D.F. Barrow, "Infinite exponentials"
Barrow coined E^n_{k=1} a_k for arbitrary nested exponentials, or
alternatively E(a1, a2, a3, ..., an) = a1^(a2^(a3^...^an)).
Shell uses Barrow's notation plus E^n_{k=1} (a_k; z) for
what Barrow calls "margerin functions". This is similar to
Galidakis' notation ^y(a, z) for iterated exponential functions.
Thron uses T_n(z) for what Barrow calls "margerin functions".
Brunson uses T(a1, a2, a3, ..., an) for nested exponentials.
Instead of Barrow's "E", there are 2 other logical choices:
"E", Rationale: Used by several authors: Barrow and Shell. It is the
original letter used for n-ary exponential / n-ary tower notations.
It is the Greek letter associated with the first letter in "Exponential".
Disadvantages: "E" looks like "Sigma". Also, if we are going to call
this "n-ary tower" notation, then "E" is not in line with tradition.
"T", Rationale: Used by 2 authors: Thron and Brunson. If we are
going to call the RESULT of nested exponentials "towers", then
by the tradition of "n-ary Sums" and "n-ary Products" we should
call these "n-ary Towers" which implies "T" instead of "E".
- E ...
- T ...
- (other)
whatever letter is used, lets say [?],
it should be usable in the following cases:
* [?]_{k=1}^n a_k
* [?]_{k=1}^n(a_k; z)
* [?]_{k=1}^n(a_k; a_n+1, z)
* [?](a1, a2, a3, ... an)
* [?](a1, a2, a3, ...; z)
* [?](avector) where avector = (a1, a2, a3, ...)
* [?]^n(x; z) = "iterated exponentials"
* [?]^n(x) = "tetration"