A characterization of sets and operations invariant under bijections
The authors call an operation (here binary, in the paper mostly \(n\)-ary) \(F: [0,1]\times[0,1]\to[0,1]\) invariant (or selfconjugate) if, for every bijection \(f\) from \([0,1]\) onto \([0,1]\) the equation \(F[f(x),f(y)]=f[F(x,y)]\) \((x,y\in[0,1])\) holds. The following is one result (the last) and the end of the paper. There exist exactly two increasing (i.e. nondecreasing in each variable) invariant binary operations: \textit{min} and the one with \(F(x,1)=x\), \(F(1,y)=y\) and \(F\) zero everywhere else on \([0,1]^2.\) ``More detailed considerations of this rich family of invariant binary operations are left to other papers''.
invariant (selfconjugate) operations, invariant domains
scientific article; zbMATH DE number 2113890