## String Algebra

Given a group G of order g, let U denote the set of all ordered multi-
sets of elements of G. Elements of U are called "strings", and the
number of components in a string q is called the "length" of q. A
string consisting only of t repetitions of the identity element of G is
signified by I_t and is called a "null string". Let V denote the subset
of U consisting of multi-sets with exactly g components, and let A
denote the subset of V consisting of multi-sets of exactly g DISTINCT
components.
For any string q of length t, and any integer d that divides t, we
define the "contraction" C(q,d) as the ordered set of elements of G
given by cumulative compositions of the components of q taken in t/n
sets of n consecutive components. For example, if q consists of
{q1, q2, q3, q4, q5, q6} then C(q,2) is defined as {q1q2, q3q4, q5q6}.
Also for any string q and any integer k we define the "rotation"
R(q,k) as the string consisting of the components of q all shifted k
places to the left, wrapping the leftmost element to the rightmost
place with each shift. Thus, with q as above, the string R(q,2) is
given by {q3, q4, q5, q6, q1, q2}.
The "successor" of any string q of length t is defined as the string
S(q) = {C(R(q,k),t), k=0,1,..,t}
We will denote j iterations of the successor function beginning with q
as S(q,j).
These strings of group elements have many interesting algebraic
possibilities with various operations. For example, given two
strings
p = {p1, p2,...pj} q = {q1, q2, ...,qk}
of length j and k respectively, we can define a string of length k + j
by the catenation
p + q = {p1, p2,..., pj, q1, q2, ..., qk}
and a string of length kj by the compositional "cross product"
p x q = {p1q1,p1q2,..,p1qk,p2q1,p2q2,..,p2qk,..,pjq1,pjq2,..,pjqk}
If k = j we also have the "interleaving" operation
p # q = {p1, q1, p1, q2, ..., pk, qk}
and the "dot product" defined by
pq = {p1q1, p2q2, ..., pkqk}
These operations, together with the contraction, rotation, and
successor functions defined above, produce a very interesting
algebraic structure.

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