単語のすべての順列の集合のような言語演算子の帰納的定義は、シャッフル演算子から来た

Let $X$ be a finite alphabet. Given two words $u, v in
X^{ast}$ the shuffle operator is defined to be $$ u || v := { u_1
v_1 u_2 v_2 ldots u_n v_n : 1 le i le n, u_i, v_i in X^{ast},
u = u_1 ldots u_n, v = v_1 ldots v_n } $$ i.e. it is the set of
all “interlaced” strings (notice its signature $X^{ast}times
X^{ast}to 2^{X^{ast}}$). For example $$ ab || c = { cab, acb,
abc }. $$ There is also an inductive definition begin{align*}
varepsilon || u & = u || varepsilon = {u} \ au || bv & = a(u
|| bv) cup b(au || v). end{align*} This operator could be
extended to languages. Now, for a language $L subseteq X^{ast}$
denote by $mbox{perm}(L)$ the set of all permutations of words
from $L$, for example $mbox{perm}({ abc }) = { abc, bac, acb,
cba, cab, bca }$. Now we have the inductive definition for $x in
X$ and $u in X^{ast}$ begin{align*} mbox{perm}(varepsilon) & =
{varepsilon} \ mbox{perm}(xu) & = x || mbox{perm}(u).
end{align*} which defines it for words, and then extended to
languages. Now instead of all permutations, just consider the
rotations, or the cyclic permutations, i.e. for $u in X^{ast}$ $$
mbox{cycle}(L) := { vu : uv in L}. $$ For example
$mbox{cycle}({abc}) = {abc, bca, cab}$. And similar the
rotations and reflections, where if we define $mbox{reverse}(u) =
u_n ldots u_1$ for $u = u_1 ldots u_n$ with $u_i in X$, then for
$L subset X^{ast}$ the smallest set, called $mbox{di}(L)$ such
that $$ u in mbox{di}(L) Rightarrow mbox{cycle}({u})subseteq
mbox{di}(L), quadmbox{and}quadmbox{reverse}(u) in
mbox{di}(L). $$ For example $mbox{di}({abcd}) = {abcd, bcda,
cdab, dabc, cbad, dcba, adcb, badc }$. Now do the
$mbox{cycle}$-operator and the $mbox{di}$-operator admit a
similar inductive definition in terms of some composition of words,
i.e. exists there some composition such that we can define them
inductively like $mbox{perm}$?

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