Diagonal mappings of set-theoretic solutions of the Yang-Baxter equation

It is well known that, for involutive non-degenerate solutions of the Yang-Baxter equation, the operations $T(x)=\tau_x^{-1}(x)$ and $U(x)=\sigma_x^{-1}(x)$ are mutually inverse. We study these mappings in the non-involutive case, under different additional conditions (like bijectivity, non-degeneracy, left non-degeneracy). We also provide axiomatic description of multipermutation solutions which enables one to find a different shape of the diagonal mappings.