Some algebraic and analytical properties of a class of two-place functions

This article deals with the formula $f^{(-1)}\left(F\right(f\left(x\right),f\left(y\right)\left)\right)$ generated by a one-place function $f:[0,1]\to [0,1]$ and a binary function $F:{[0,1]}^2\to [0,1]$. When the f is a strictly increasing function and F is a continuous, non-decreasing and associative function with neutral element in $[0,1]$, the following algebraic and analytical properties of the formula are studied: idempotent elements, the continuity (resp. left-continuity/right-continuity), the associativity and the limit property. Relationship among these properties is investigated. Some necessary conditions and some sufficient conditions are given for the formula being a triangular norm (resp. triangular conorm). In particular, a necessary and sufficient condition are expressed for obtaining a continuous Archimedean triangular norm (resp. triangular conorm). When the f is a non-decreasing surjective function and F is a non-decreasing associative function with neutral element in $[0,1]$, we investigate the associativity of the formula.