A Generalized Monoid of Words with Applications to Divergent Arithmetic Products
A generalized monoid of words is constructed as an extension of the free monoid Sigma-star with elements of controlled infinite length. The construction uses a bidirectional prefix-suffix metric and an asymptotic equivalence relation on moderate nets of finite words. The resulting quotient is a monoid carrying a natural partial order, a length homomorphism, and a well-defined reversal involution. Moulds, in the sense of Ecalle's resurgent analysis, are defined on this monoid. The logarithmic window provided by the asymptotic equivalence guarantees that moulds depending only on logarithmic prefixes descend to well-defined functionals on the quotient. The framework is applied to the regularization of divergent arithmetic products whose oscillations follow a regular pattern. The alternating products of integers, primes, and factorials acquire canonical finite values that coincide with zeta regularization. A symmetrized functional cancels leading oscillations, and a logarithmic Cesaro renormalization extracts the constant term. The method is then extended to products beyond the reach of classical regularization, such as products whose sign sequences are constant on dyadic blocks. A conjecture is proposed for the Thue-Morse product. The selection of evaluation functionals and renormalization schemes is systematized according to the divergence type of the arithmetic sequence.