(Ä-operation) in (Fm, FΔm) representations Å-operation in (Fm, FΔm) representations 4) $\begin{array}{c}{F}_{m}\left({a}_{i}\otimes {a}_{j}\right)=〈m\left({a}_{i},{a}_{j}\right)〉\\ =〈m\left({a}_{i}\right),m\left({a}_{j}\right)〉\end{array}$ ${F}_{m}\left({a}_{i}\otimes {a}_{i}\right)={F}_{m}\left({a}_{i}\right)=m\left({a}_{i}\right)$ 1) $\begin{array}{c}{F}_{m}\left({a}_{i}\oplus {a}_{j}\right)={F}_{m}\left({a}_{i}\right)+{F}_{m}\left({a}_{j}\right)\\ =m\left({a}_{i}\right)+m\left({a}_{j}\right)\end{array}$ $\begin{array}{c}{F}_{m}\left({a}_{i}\oplus {a}_{i}\right)={F}_{m}\left({a}_{i}\right)+{F}_{m}\left({a}_{i}\right)\\ =m\left({a}_{i}\right)+m\left({a}_{i}\right)=2m\left({a}_{i}\right)\end{array}$ 2) ${F}_{\Delta m}\left({a}_{i}\otimes {a}_{j}\right)=0$ ${F}_{\Delta m}{\left({a}_{i}\otimes {a}_{j}\right)}_{\oplus }\ne 0$ with ${\left({a}_{i}\otimes {a}_{j}\right)}_{\oplus }=\left({a}_{i}\otimes {a}_{j}\right)\oplus {a}_{k}$ 2) $\begin{array}{l}{F}_{\Delta m}\left(a\oplus b\oplus c\right)\\ ={F}_{\Delta m}\left(a\right)+{F}_{\Delta m}\left(b\right)+{F}_{\Delta m}\left(c\right)+{F}_{\Delta m}{\left(a\underset{_}{\oplus }b\right)}_{ab}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{F}_{\Delta m}{\left(a\underset{_}{\oplus }c\right)}_{ac}+{F}_{\Delta m}{\left(b\underset{_}{\oplus }c\right)}_{bc}\end{array}$ 3) ${F}_{\Delta m}{\left({a}_{i}\otimes {a}_{j}\right)}_{\oplus }={F}_{\Delta m}\left({a}_{i}\right)$ ${F}_{\Delta m}\left(\left({a}_{i}\otimes {a}_{j}\right)\oplus {a}_{k}\right)={F}_{\Delta m}\left(\left({a}_{i}\right)\oplus {a}_{k}\right)$ 3) $\begin{array}{l}{F}_{\Delta m}\left[\left({a}_{i}\oplus {a}_{j}\right)\oplus {a}_{k}\right]\\ ={F}_{\Delta m}\left[\left({a}_{i}\oplus {a}_{k}\right)\oplus \left({a}_{j}\oplus {a}_{k}\right)\right]\end{array}$ 4) $\begin{array}{l}{F}_{m}\left[\left({a}_{i1}\otimes {a}_{j2}\right)\oplus \left({a}_{i2}\otimes {a}_{j1}\right)\right]\\ ={F}_{m}\left[2{\left({a}_{i}\otimes {a}_{j}\right)}_{12}\right]\end{array}$ 4) $\begin{array}{l}{F}_{\Delta m}\left[\left({a}_{i}\otimes {a}_{j}\right)\oplus \left({a}_{k}\otimes {a}_{r}\right)\right]\\ ={F}_{\Delta m}\left[\left({a}_{i}\otimes {a}_{j}\right)\oplus \left({a}_{k}\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{F}_{\Delta m}\left[\left({a}_{i}\otimes {a}_{j}\right)\oplus \left({a}_{r}\right)\right]\end{array}$ 5) ${F}_{\Delta m}\left(2{a}_{i}\otimes {a}_{j}\right)={F}_{\Delta m}\left(2\left({a}_{i}\otimes {a}_{j}\right)\right)$ $\begin{array}{l}{F}_{\Delta m}\left(n\left({a}_{i}\otimes {a}_{j}\right)\oplus m\left({a}_{k}\otimes {a}_{r}\right)\right)\\ =\left(n×m\right){F}_{\Delta m}\left(\left({a}_{i}\otimes {a}_{j}\right)\oplus \left({a}_{k}\otimes {a}_{r}\right)\right)\end{array}$ 6) $\begin{array}{l}{F}_{\Delta m}\left[{a}_{i}\oplus {a}_{j}\right]+{F}_{\Delta m}\left[{a}_{i}\oplus {a}_{k}\right]\\ ={F}_{\Delta m}\left[{a}_{i}\oplus \left({a}_{j}+{a}_{k}\right)\right]\end{array}$