Recognizing Numbers
Pranshu Gaba, Arnab Sur
TL;DR
This work investigates recognizability by finite monoids for additive and multiplicative number-based monoids across $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$. By analyzing morphisms to finite monoids and exploiting properties such as arbitrary divisibility, it characterizes recognizable sets for several additive monoids (e.g., $\mathbb{Z}_{\ge 0}$ yields ultimately periodic sets, $\mathbb{Z}$ yields periodic sets, while $\mathbb{Q}_{\ge 0}$ and boolean combinations collapse to a few trivial families; additive groups $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ are even more restrictive). In the multiplicative setting, after reducing to zero-free cases and applying direct-product decompositions, the results show that the positive parts of arbitrarily divisible monoids have trivial recognizability, while infinitely generated free-monoid cases (e.g., $\mathbb{N}^*$ under multiplication) admit uncountably many recognizable subsets and many natural properties remain non-recognizable. Overall, recognizability by finite monoids appears inadequate to capture natural structure for these number-based monoids, motivating exploration of alternative algebraic frameworks such as matrices or relational models for timed or relational aspects.
Abstract
The use of monoids in the study of word languages recognized by finite-state automata has been quite fruitful. In this work, we look at the same idea of "recognizability by finite monoids" for other monoids. In particular, we attempt to characterize recognizable subsets of various additive and multiplicative monoids over integers, rationals, reals, and complex numbers. While these recognizable sets satisfy properties such as closure under Boolean operations and inverse morphisms, they do not enjoy many of the nice properties that recognizable word languages do.