## FANDOM

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Recently, we've been having some discussion of Bowers' pentational arrays.  Bowers talks about pentational arrays briefly, but specific details are lacking.  Here is my attempt at a rigorous definition of pentational array notation.

## Definition of pentational structures

We define the set $$PE$$ of pentational structure expressions as follows.

$$PE$$ is the smallest set such that

0, 1, X are in $$PE$$. If f and g are in $$PE$$, then f + g, f * g, $$X^f$$, and $$X \uparrow \uparrow f$$ are in $$PE$$.

Note that different expressions for pentational structures can express the same pentational structure.

We define the equivalence relation ~ as follows.  Given that f and g are in $$PE$$:

f + g ~ g + f

f + 0 ~ f

f * g ~ g * f

f * 0 ~ 0

f * 1 ~ f

$$X^0 \sim 1$$

$$X^1 \sim X$$

$$X\uparrow\uparrow 0 \sim 1$$

$$X \uparrow\uparrow 1 \sim X$$

$$X^{f + g} \sim (X^f) * (X^g)$$

$$X^{X\uparrow \uparrow f} \sim X \uparrow\uparrow (f+1)$$

~ is the smallest equivalence relation that satisfies the above expressions.

The set $$P$$ of pentational structures is defined as the set $$PE$$ modulo ~.  That is, $$P$$ is the set of pentational structure expressions where expressions that are related by ~ are considered the same structure.

## Ordering on pentational structures

We need to order the pentational structures.  For a pentational structure f, define N(f) to be the minimum number of symbols in an expression for f.  We will define the comparsion between f and g by induction on max {N(f), N(g)}.

Case 1: 0 < g is true for all g not equal to 0.

Case 2: 1 < g is true for all g not equal to 0 or 1.

Case 3: f < 0 is false for all f.

Case 4: f < 1 is false unless f = 0.

Case 5: At least one of f or g is the sum of two or more expressions.

Let $$f = f_1 + f_2 + \ldots + f_m$$ where $$f_1 \ge f_2 \ge \ldots \ge f_m$$. (Note that \N(f_i) < \N(f)\) so we can assume by induction that comparisons for the $$f_i$$ have already been defined.)

Let $$g = g_1 + g_2 + \ldots + g_n$$ where $$g_1 \ge g_2 \ge \ldots \ge g_n$$.

$$f < g \Leftrightarrow \exists i (f_i < g_i \wedge \forall j < i (f_j = g_j)) \vee (\forall i \le m (f_i = g_i) \wedge n > m)$$.

(In other words, we compare the summands term by term, starting from the largest, until we find a pair that are different.  Whichever summand is greater will belong to the greater sum.)

Case 6: At least one of f or g can be written as the product of two or more expressions.

Let $$f = f_1 * f_2 * \ldots * f_m$$ where $$f_1 \ge f_2 \ge \ldots \ge f_m$$, and no $$f_i)$$ is of the form $$X^{a+b}$$ where a and b are nonzero.

Let $$g = g_1 * g_2 * \ldots * g_n$$ where $$g_1 \ge g_2 \ge \ldots \ge g_n$$, and no $$g_i)$$ is of the form $$X^{a+b}$$ where a and b are nonzero.

Then as before,

$$f < g \Leftrightarrow \exists i (f_i < g_i \wedge \forall j < i (f_j = g_j)) \vee (\forall i \le m (f_i = g_i) \wedge n > m)$$.

Case 7: $$f = X^F, g = X^G$$

$$f < g \Leftrightarrow F < G$$.

Case 8: $$f = X \uparrow\uparrow F, g = X \uparrow\uparrow G$$

$$f < g \Leftrightarrow F < G$$.

Case 9: $$f = X^F, g = X \uparrow\uparrow G$$

If $$F = X \uparrow \uparrow H$$, then $$f < g \Leftrightarrow H+1 < G$$. Otherwise, $$f < g \Leftrightarrow F < g$$.

Case 10: $$f = X \uparrow\uparrow F, g = X^G$$

If $$G = X \uparrow \uparrow H$$, then $$f < g \Leftrightarrow F < H+1$$. Otherwise, $$f < g \leftrightarrow f < G$$.

This completely defines the comparison relation.

## Standard form for pentational structures

We define the standard form for a pentational array f.

If f is a nontrivial sum, order the summands from largest to smallest, and express each summand in standard form.

If f is of the form $$X^{g+h}$$ express it in the form $$(X^g) * (X^h)$$, and express g and h in standard form.

If f is a nontrivial product, order the factors from largest to smallest, and express each factor in standard form.

If f is of the form $$X^{X\uparrow\uparrow f}$$, express it in the form $$X \uparrow \uparrow (f+1)$$, with f expressed in standard form.

## Fundamental sequences for pentational structures

Given f expressed in standard form, define f[n], the nth element of the fundamental sequence for f, as follows:

If $$f = 0, f[n] = 0$$.

If $$f = g+1, f[n] = g$$.

If $$f = f_1 + f_2 + \ldots + f_m$$, then $$f[n] = f_1 + f_2 + \ldots + f_m[n]$$.

If $$f = f_1 * f_2 * \ldots * f_m$$, then $$f[n] = f_1 * f_2 * \ldots * f_m[n]$$.

If $$f = X^1$$, $$f[n] = n$$.

If $$f = X^g$$, $$f[n] = X^{g[n]}$$.

If $$f = X\uparrow\uparrow {g + 1}, f[n] = (X\uparrow\uparrow g)*(X\uparrow\uparrow g)* \ldots *(X\uparrow\uparrow g)$$, where there are n terms in the product.

If $$f = X\uparrow\uparrow g$$ and g is not of the form h+1, then $$f[n] = X\uparrow\uparrow (g[n])$$.

## Prime blocks for pentational structures

We will use the same definition for the prime blocks as used by FB100Z in his Ordinal BEAF notation.

Call f a successor pentational structure if f is of the form g+1 for some g.

Call f a limit pentational structure if f is neither a successor pentational structure nor 0.

Define $$P_p (\alpha)$$, the prime block of $$\alpha$$, as follows:

$$P_p(0) = \lbrace \rbrace$$.

$$P_p(f + 1) = \lbrace f \rbrace \cup P_p (f)$$.

If f is a limit pentational structure, $$P_p (f) = P_p (f[p])$$.

## Pentational Arrays

Having defined prime blocks for pentational structures, we are now ready to define pentational arrays.  A pentational array is a map from pentational structures to the natural numbers; we can notate this as

$$\lbrace (\alpha_1, n_1), (\alpha_2, n_2), \ldots, (\alpha_m, n_m) \rbrace$$, where the $$\alpha_i$$ are increasing.  The 0 structure will map to the base number, and the 1 structure will map to the prime number.  The pilot P will be the smallest structure greater than 1 mapping to a positive number, and the copilot CP will be the structure such that P = CP+1, if such a structure exists.  The definition of BEAF is the usual one.

Hopefully there are no major mistakes, and we have successfully defined pentational arrays.

## Relationship between structures and ordinals

In the following, A will stand for an arbitrary structure with corresponding ordinal $$\alpha$$.

Structure Ordinal $$X \uparrow\uparrow X$$ $$\varepsilon_0$$ $$X \uparrow\uparrow X + 1$$ $$\varepsilon_0 + 1$$ $$X \uparrow\uparrow X + A$$ $$\varepsilon_0 + \alpha$$ $$X \uparrow\uparrow X * 2$$ $$\varepsilon_0 * 2$$ $$X \uparrow\uparrow X * A$$ $$\varepsilon_0 * \alpha$$ $$(X \uparrow\uparrow X) ^ 2$$ $$\varepsilon_0 ^ 2$$ $$(X \uparrow\uparrow X) ^ n$$ $$\varepsilon_0 ^ n$$ $$X \uparrow\uparrow (X + 1)$$ $$\varepsilon_0 ^ {\omega}$$ $$X ^ {X \uparrow\uparrow X}$$ $$\omega ^ {\varepsilon_0 * \omega}$$ $$X ^ {X \uparrow\uparrow X} + A$$ $$\omega ^ {\varepsilon_0 * \omega} + \alpha$$ $$X ^ {X \uparrow\uparrow X} * 2$$ $$\omega ^ {\varepsilon_0 * \omega} * 2$$ $$X ^ {X \uparrow\uparrow X} * A$$ $$\omega ^ {\varepsilon_0 * \omega} * \alpha$$ $$X ^ {X \uparrow\uparrow X + 1}$$ $$\omega ^ {\varepsilon_0 * \omega + 1}$$ $$X ^ {X \uparrow\uparrow X + A}$$ $$\omega ^ {\varepsilon_0 * \omega + \alpha}$$ $$X ^ {X \uparrow\uparrow X} * X \uparrow\uparrow X$$ $$\omega ^ {\varepsilon_0 * (\omega + 1)}$$ $$X ^ {X \uparrow\uparrow X} * X \uparrow\uparrow X * A$$ $$\omega ^ {\varepsilon_0 * (\omega + 1 + \alpha)}$$ $$X ^ {X \uparrow\uparrow X} * (X \uparrow\uparrow X)^2$$ $$\omega ^ {\varepsilon_0 ^ 2 }$$ $$X ^ {X \uparrow\uparrow X} * (X \uparrow\uparrow X)^n$$ $$\omega ^ {\varepsilon_0 ^ n }$$ $$X ^ {X \uparrow\uparrow X * 2}$$ $$\omega ^ {\varepsilon_0 ^ {\omega} }$$ $$X ^ {X \uparrow\uparrow X * (1 + A}$$ $$\omega ^ {\varepsilon_0 ^ {\omega * \alpha} }$$ $$X ^ {(X \uparrow\uparrow X)^2}$$ $$\omega ^ {\varepsilon_0 ^ {\varepsilon_0} }$$ $$X ^ {(X \uparrow\uparrow X)^n}$$ $$\omega ^ {\omega ^ {\varepsilon_0 * n} }$$ $$X ^ {X ^ {X \uparrow\uparrow X}}$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega}}$$ $$X ^ {X ^ {X \uparrow\uparrow X}} + A$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega}} + \alpha$$ $$X ^ {X ^ {X \uparrow\uparrow X}} * A$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega}} * \alpha$$ $$(X ^ {X ^ {X \uparrow\uparrow X}}) ^ n$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega} * n}$$ $$(X ^ {X ^ {X \uparrow\uparrow X + 1}})$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega + 1}}$$ $$(X ^ {X ^ {X \uparrow\uparrow X + A}})$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega + \alpha}}$$ $$(X ^ {X ^ {X \uparrow\uparrow X} * X \uparrow \uparrow X})$$ $$\omega ^ {\omega ^{\varepsilon_0 * \omega + \varepsilon_0}}$$ $$(X ^ {X ^ {X \uparrow\uparrow X} * (X \uparrow \uparrow X)^n})$$ $$\omega ^ {\omega ^{\varepsilon_0 ^ n}}$$ $$(X ^ {X ^ {X \uparrow\uparrow X * 2} })$$ $$\omega ^ {\omega ^{\varepsilon_0 ^ {\omega} }}$$ $$(X ^ {X ^ {X \uparrow\uparrow X * ( 1 + A)} })$$ $$\omega ^ {\omega ^{\varepsilon_0 ^ {\omega * \alpha}}}$$ $$(X ^ {X ^ {X \uparrow\uparrow X * A} })$$ $$\omega ^ {\omega ^{\varepsilon_0 * (\omega * \alpha)}}$$ $$X ^ {X ^ {(X \uparrow\uparrow X)^2}}$$ $$\omega ^ {\omega ^ {\varepsilon_0 ^ {\varepsilon_0} } }$$ $$X ^ {X ^ {(X \uparrow\uparrow X)^n}}$$ $$\omega ^ {\omega ^ {\omega ^ {\varepsilon_0 * n} } }$$ $$X ^ {X ^ {X ^ {X \uparrow\uparrow X}}}$$ $$\omega ^ {\omega ^ {\omega ^{\varepsilon_0 * \omega}}}$$ $$X \uparrow \uparrow (2X)$$ $$\varepsilon_1$$ $$X \uparrow \uparrow ((A+1) X)$$ $$\varepsilon_(\alpha)$$ $$X \uparrow \uparrow (X \uparrow \uparrow X)$$ $$\varepsilon_{\varepsilon_0}$$ $$X \uparrow \uparrow (X \uparrow \uparrow (X \uparrow\uparrow X))$$ $$\varepsilon_{\varepsilon_{\varepsilon_0}}$$ $$X \uparrow \uparrow \uparrow X$$ $$\phi_2 (0)$$