I wonder what kinds of infinite products of matrices, elements of Banach algebras, and complex numbers emerge from rankings in rank.
Suppose that $ lambda $ is a cardinal and $ j_ {1}, dots, j_ {k}: V _ { lambda} rightarrow V _ { lambda} $ are not trivial elementary embeddings. To let $ mathrm {crit} _ {n} (j_ {1}, dots, j_ {k}) $ be that $ n $th element in the sentence $ { mathrm {crit} (j) midj in langle j_ {1}, dots, j_ {k} rangle } $,
To let $ p_ {n, j_ {1}, dots, j_ {k}} (x_ {1}, dots, x_ {k}) $ denote the defined noncommutative polynomial
$$ 1 + sum {x_ {a_ {1}} dots x_ {a_ {s}} mid mathrm {crit} {j_ {a_ {1}} * dots * j_ {a_ {s}} )
= mathrm {crit} _ {n} (j_ {1}, dots, j_ {k}), $$
$$ mathrm {critical} (j_ {a_ {1}} * dots * j_ {a_ {r}}) < mathrm {critical} _ {n} (j_ {1}, dots, j_ {k} )) , text {for all} , 1 leq r <s }. $$
If the elementary embeddings $ j_ {1}, dots, j_ {k} $ are unique, then we will write $ p_ {n} (x_ {1}, dots, x_ {k}) $ to the $ p_ {n, j_ {1}, dots, j_ {k}} (x_ {1}, dots, x_ {k}) $,
The variables $ x_ {1}, dots, x_ {k} $ do not commute with each other (like that $ x_ {1}, dots, x_ {k} $ should be considered as matrices of elements of a Banach algebra.
The polynomials $ p_ {n} (x_ {1}, dots, x_ {k}) $ fulfill the infinite product formula
$$ lim_ {n rightarrow infty} p_ {n} (x_ {1}, dots, x_ {k}) cdot dots cdot p_ {0} (x_ {1}, dots, x_ { k}) = frac {1} {1 (x_ {1} + dots + x_ {k})}. $$
For example when $ j_ {1} = dots = j_ {k} $, then $ p_ {n} (x_ {1}, dots, x_ {k}) = 1+ (x_ {1} + dots + x_ {k}) ^ {2 ^ {n}} $ for all $ n in omega. $
Can anyone give a nontrivial example of a sequence of different nontrivial elementary embeddings $ j_ {1}, dots, j_ {k} $ along with $ r times r $ matrices $ A_ {1}, dots, A_ {k} $ where if $ B_ {n} = p_ {n} (A_ {1}, dots, A_ {k}), $ then

$ 1 (A_ {1} + dots + A_ {k}) $ is not singular,

There is a closed expression for the sequence $ (B_ {n}) _ {n in omega} $ (especially if every entry in each $ A_ {i} $ is algebraically over $ mathbb {Q} $, then the coefficients in $ B_ {n} $ should be calculable at least in polynomial time),

$$ lim_ {n rightarrow infty} B_ {n} cdot dots cdot B_ {0} = frac {1} {1 (A_ {1} + dots + A_ {k})}, $$

If $ alpha = mathrm {crit} (j_ {i}) $ for some $ i $, then $$ sum {A_ {i} mid 1 leq i leqk, mathrm {crit} (j_ {i}) = alpha } $$ is not singular and

The sequence $ (B_ {n}) _ {n in omega} $ is not identical after all $ 1. $
I hope that conditions 4 and 5 rule out all trivial cases.
I am also interested in some generalizations of this question and am pleased about answers to the general questions. One way to generalize polynomials, for example, is to use an infinite number of variables $ x_ {r} $ corresponding to infinitely many elementary embeddings. Another generalization would be the choice $ A_ {1}, dots, A_ {k} $ from some banach algebra (or another room) or for $ A_ {1}, dots, A_ {k} $ simply be complex numbers. Another way to generalize this question is to use algebraic structures that resemble rankinrank embedding in terms of critical points, a composition operation, and so forth, but that actually arise in the algebras of elementary embedding.
Some of my computer calculations indicate that there are these nontrivial infinite products, including the following candidate to answer this question.
Suppose that $ k = 2 $ and $ j: V _ { lambda} rightarrow V _ { lambda} $, To let $ j_ {1} = j * j, j_ {2} = j * j * j $, Then the following sequence is the sequence $ (p_ {0} (x, ix), dots, p_ {15} (x, ix)) $:
$ (i cdot x + 1, x + 1, 1, i cdot x ^ 2 + 1, x ^ 4x ^ 3 + 1, i cdot x ^ 3 + 1, x ^ 7 + 1, (i) cdot x ^ 6 + 1, 1, 1, 1, 1, 1, 1, 1) $,
The value of the polynomial $ p_ {16} (x, ix) $ is unknown, and I do not know if the order of polynomials $ (p_ {n} (x, ix)) _ {n} $ is in some closed form or can be calculated in polynomial from $ n $ Time. The expression for $ (p_ {0} (x, y), dots, p_ {15} (x, y)) $ is 33982 characters long, hence the friendliness of the polynomials $ (p_ {0} (x, ix), dots, p_ {15} (x, ix)) $ is pretty unusual.
We also have $ (p_ {0, j, j * j} (i, 1), points, p_ {11, j, j * j} (i, 1)) = (1 + i, 1, 1 + i, 1 ) 0, 1, 1, 1, 1, 1, 1), $ and
$ (p_ {0, j * j, j * (j * j) * j} (1, i), dots, p_ {15, j * j, j * (j * j) * j} (1, i)) =
(1 + i, 1, 1, 1 + i, 1 + i, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1) $
One should be careful, and Laver tables should never be assumed to continue a pattern, as the Laver tables are filled with temporary patterns and even some longlived patterns that must end up under great cardinal hypotheses.