  Let $X$ be a nonempty set and let $i,j \in \{1,2,3,4\}$. We say that a binary
operation $F:X^2\to X$ is $(i,j)$-selective if $$
F(F(x_1,x_2),F(x_3,x_4))~=~F(x_i,x_j), $$ for all $x_1,x_2,x_3,x_4\in X$. In
this paper we provide characterizations of the class of $(i,j)$-selective
operations. We also investigate some subclasses by adding algebraic properties
such as associativity or bisymmetry.
