Markov–Kakutani fixed-point theorem

Büchi arithmetic of base k is the first-order theory of the natural numbers with addition and the function $V_k(x)$ which is defined as the bigger power of k dividing x, named in honor of the Swiss mathematician Julius Richard Büchi. The signature of Presburger arithmetic contains only the addition operation, $V_k$ and equality, omitting the multiplication operation entirely.

Unlike Peano arithmetic, Büchi arithmetic is a decidable theory. This means it is possible to effectively determine, for any sentence in the language of Büchi arithmetic, whether that sentence is provable from the axioms of Presburger arithmetic.

Büchi arithmetic and automaton

A subset $X\subseteq {\mathbb{\mathbb{N}}}^n$ is definable in Bûchi arithmetic of base k if and only if it's ''k''-recognisable.

If $n=1$ this means that the set of integers of X in base k is accepted by an automaton. Similarly if $n>1$ there exists an automata that read firsts digit, then second digits, and so on, of n integers in base k, and accept the words if the n integers are in the relation X.

Properties of Büchi arithmetic

Two numbers k and l are multiplicatively dependant if there exists integers n and m such that $k^n=l^m$.

If k and l are multiplicatively dependant then Büchi arithmetic of base k and l have the same expressivity. Indeed $V_l$ can be defined in $\text{FO}(V_k,+)$.

Else the theory with both $V_k$ and $V_l$ function is equivalent to Peano arithmetic the logic with addition and multiplication since multiplication is definable in $\text{FO}(V_k,V_l,+)$.

On the other hand by Semenov's theorem, if a relation is definable in both k and l Büchi arithmetics it is definable in Presburger arithmetic.