Classification of admissible Stachowiak operator pairs

Characterize all pairs of functions and operators $(M,f)$ satisfying the stated Stachowiak framework that define EML-type operators.

Background

The paper presents a general scheme in which an EML-type operator is written as $S(x,y)=M(f(x),f^{-1}(y))$, subject to algebraic conditions on the noncommutative operator $M$, together with a distinguished neutral element and constant. Known EML, EDL, LDE, PLI, and PLM operators fit this framework.

Despite this unifying representation, the authors state that the admissible choices of $M$ and $f$ have not been characterized. The open problem concerns a general classification of all such pairs, rather than the verification of any particular known operator.