An Analysis of Cascade Chains in the Stern-Brocot Sequence

Abstract

We introduce and analyze the algebraic framework of finite cascade sequences, defined as
deterministic paths terminating in a unique absorbing state [1][2]. Three complementary perspectives are
developed: graph-theoretic, probabilistic, and operator-theoretic. In the graph view, cascade sequences
form directed acyclic graphs with a terminal sink [3][4]. In the Markov chain formulation, they
correspond to absorbing chains with deterministic absorption time [7][8][13]. In the operator view, the
transition operator T is shown to be nilpotent, generating a finite algebra with minimal polynomial
xk[9][15].
Further the semi group {T,T2,…,Tk} under composition, the monoid obtained by adjoining the identity,
and the absorbing element Tk are characterized [5][6][11][12]. These are shown to be isomorphic to
truncated polynomial semigroups and monoids. Extending the algebraic cascade property, we prove that
each finite cascade sequence is a cyclic module over the truncated polynomial ring Q[x]/(xk), with
dimension k over Q [9].
Together, these results unify combinatorial, probabilistic, and algebraic aspects of cascade sequences,
establishing a rigorous foundation for their study and connecting them to classical structures in semigroup
theory, ring theory, and module theory [2][5][6].