Relative Prime Factorization and Finite-State Presentations under Fixed Finite-Monoid Observation
Let $L\subseteqΣ^*$ and fix a morphism $h:Σ^*\to M$ into a finite monoid. We study exact factorization and canonical presentation in the relative syntactic congruence $θ_{L,h}:=\equiv_L\cap\ker h$. We separate unique factorization from finite direct presentation. An exhaustively computer-checked $36$-element quotient has a unique exact prime factorization for every live non-unit class, yet its valid prime-return rules contain an infinite family, so unique factorization does not imply the finite relative presentation property (FRP), even for a finite quotient. We lift the same defect to a nonregular context-free language with an infinite relative quotient and finite prime spectrum. To isol...
| Authors | Takayuki Kuriyama |
|---|---|
| Primary category | cs.FL |
| Published | 3 September 2026 |
| Last updated at source | 3 September 2026 |
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