arXiv subjects

Didier Caucal

Publications and source records attributed to Didier Caucal.

4 recordsLinked to original sources

arXiv2022

Synchronizable functions on integers

For all natural numbers a,b and d > 0, we consider the function f_{a,b,d} which associates n/d to any integer n when it is a multiple of d, and an + b otherwise; in particular f_{3,1,2} is the Collatz function. Coding in base a > 1 with b < a, we realize these functions by input-deterministic letter-to-letter transducers with additional output final words. This particular form allows to explicit, for any integer n, the composition n times of such a transducer to compute f^n_{a,b,d}. We even realize the closure under composition f^*_{a,b,d by an infinite input-deterministic letter-to-letter transducer with a regular set of initial states and a length recurrent terminal function.

cs.FL↗

arXiv2019

On Cayley graphs of basic algebraic structures

We present simple graph-theoretic characterizations of Cayley graphs for monoids, semigroups and groups. We extend these characterizations to commutative monoids, semilattices, and abelian groups.

cs.DM↗

arXiv2018

On Cayley graphs of algebraic structures

We present simple graph-theoretic characterizations of Cayley graphs for left-cancellative monoids, groups, left-quasigroups and quasigroups. We show that these characterizations are effective for the end-regular graphs of finite degree.

cs.DM↗

arXiv2016

Structural characterization of Cayley graphs

We show that the directed labelled Cayley graphs coincide with the rooted deterministic vertex-transitive simple graphs. The Cayley graphs are also the strongly connected deterministic simple graphs of which all vertices have the same cycle language, or just the same elementary cycle language. Under the assumption of the axiom of choice, we characterize the Cayley graphs for all group subsets as the deterministic, co-deterministic, vertex-transitive simple graphs.

cs.DM↗