An algebraic structure $ (X, *) $ is said to be self-distributing if it satisfies the identity $ x * (y * z) = (x * y) * (x * z) $.

Assume that $ X $ is a self-distributing algebra. Then the monoid with positive braid $ B_ {n} ^ {+} $ acts on $ X ^ {n} $ leaving

$ (x_ {1}, …, x_ {n}) cdot sigma_ {i} = (x_ {1}, …, x_ {i-1}, x_ {i} * x_ {i + 1}, x_ {i}, x_ {i + 2}, …, x_ {n}) $ and this action is called Hurwitz action.

Recall that an algebraic structure $ (X, *) $ is cancelable on the left if $ x * y = x * z $ involved $ y = z $. Yes $ X $ is cancelable on the left, then this action extends to a partial action of the group of braids $ B_ {n} $ sure $ X ^ {n} $ by defining

$ (x_ {1}, …, x_ {n}) cdot bc ^ {- 1} = (y_ {1}, …, y_ {n}) $ precisely when

$ (x_ {1}, …, x_ {n}) cdot b = (y_ {1}, …, y_ {n}) cdot c $, and this partial action still calls Hurwitz 's action.

Let $ mathcal {E} _ { lambda} ^ {+} $ be all of all elementary traffic jams $ j: V _ { lambda} rightarrow V _ { lambda} $. then $ mathcal {E} _ { lambda} ^ {+} $ can be endowed with a self-distributing operation $ * $ defined leaving

$ j * k = bigcup _ { alpha < lambda} j (k | _ {V _ { alpha}} $. L & # 39; operation $ * $ however, it is neither associative nor commutative. Cardinal $ lambda $ is the limit of many very great cardinals. For more information on the algebraic structure $ ( mathcal {E} _ { lambda} ^ {+}, *) $please refer to Chapter 11 of the Handbook of Set Theory. For more information on the two braids, Hurwitz's action and basic operations, refer to Chapters 1, 3 and 12 of Patrich Dehornoy's book Braids and Self-Distributivity.

Assume that $ j_ {1}, …, j_ {n} in mathcal {E} _ { lambda} ^ {+} $ and $ gamma < lambda $. Remind that $ mathrm {crit} (j) $ denotes the smallest ordinal $ alpha $ or $ j ( alpha)> alpha $. So is there only finely braids $ b in B_ {n} $ as if $ (j_ {1}, …, j_ {n}) cdot b = (k_ {1}, …, k_ {n}) $then $ mathrm {crit} (k_ {1}) < gamma, …, matthm {crit} (k_ {n}) < gamma $? If we limit ourselves to positive braids, then the answer is a yes by my answer here.