On a family of quasi-arithmetic means

Summary The main goal of this paper is to solve the idempotency equationF(x, x) = x, x 0, 1] for a class of functions of the type convex linear combination of at-norm and at-conorm. In the non-strict Archimedean case and for eachk (0, 1), we obtain a unique solutionF _k for this equation. While these functionsF _k are not associative, we also prove that they satisfy the bisymmetry condition.