A note on piecewise automorphism groups of relational systems (Q1262941)

Let \(\Omega\) be a set and R an r-place relation on \(\Omega\). An element \(g\in Sym(\Omega)\) is said to be a piecewise automorphism of (\(\Omega\),R) if there is a partition of \(\Omega\) into finitely many sets \(\Omega_ 1,...,\Omega_ t\) such that \(g| \Omega_ j\) is an R-isomorphism for all j. Let Paut(\(\Omega\),R) denote the group of piecewise automorphisms. These groups have provided useful examples, e.g. see \textit{H. Läuchli} and \textit{P. M. Neumann} [Arch. Math. Logic 27, 189-192 (1988; Zbl 0681.03030)]. We give a necessary and sufficient condition for \(Paut(\Omega,R)=Sym(\Omega)\) when \(\Omega\) is countable.