Ä-operation $\left({a}_{i},{a}_{j}\right)\in \left\{a\right\}$ Å-operation $\left({a}_{i},{a}_{j}\right)\in \left\{a\right\}$ 1) $\left({a}_{i}\otimes {a}_{i}\right)=\left(a\right)$ 1) $\left({a}_{i}\oplus {a}_{i}\right)=0$ 2) $\left({a}_{i},{a}_{j}\right)\in \left\{a\right\}$ $\left({a}_{i}\otimes {a}_{j}\right)=\left({a}_{j}\otimes {a}_{i}\right)$ 2) $\left({a}_{i}\oplus {a}_{j}\right)=\left({a}_{j}\oplus {a}_{i}\right)$ If ${\left({a}_{i}\oplus {a}_{j}\right)}_{\oplus }=\left[\left({a}_{i}\oplus {a}_{j}\right)\oplus {a}_{k}\right]$ $\to$ ${\left({a}_{i}\oplus {a}_{j}\right)}_{\oplus }\ne {\left({a}_{j}\oplus {a}_{i}\right)}_{\oplus }$ 3) $\left(n,m\right)\in N$ $\left(n{a}_{i}\otimes {a}_{j}\right)=n\left({a}_{i}\otimes {a}_{j}\right)$ $\left(n{a}_{i}\otimes m{a}_{j}\right)=\left(n×m\right)\left({a}_{i}\otimes {a}_{j}\right)$ 3) $\begin{array}{c}\left[2\left({a}_{i}\right)\oplus \left({a}_{j}\right)\right]\equiv \left[\left({a}_{i}\right)+\left({a}_{i}\right)\right]\oplus {a}_{j}\\ =\left[\left({a}_{i}\oplus {a}_{j}\right)+\left({a}_{i}\oplus {a}_{j}\right)\right]\\ =2\left({a}_{i}\oplus {a}_{j}\right)\end{array}$ 4) transitive property if $\left[\left(a\otimes b\right),\left(b\otimes c\right)\right]\to \left[\left(a\otimes c\right)\right]$ 4) transitive property if $\left[\left(a\otimes b\right),\left(b\otimes c\right),\left(a\otimes c\right)\right]$ if $\left[\left(a\oplus b\right),\left(b\oplus c\right)\right]\to \left[\left(a\oplus c\right)\right]$ 5) $\begin{array}{c}\left[\left({a}_{i}\otimes {a}_{j}\right)\otimes {a}_{k}\right]=\left[\left({a}_{i}\otimes {a}_{k}\right)\otimes {a}_{j}\right]\\ =\left[\left({a}_{k}\otimes {a}_{j}\right)\otimes {a}_{i}\right]\end{array}$ 5) $\left[\left({a}_{i}\oplus {a}_{j}\right)\oplus {a}_{k}\right]=\left[\left({a}_{i}\oplus {a}_{k}\right)\oplus \left({a}_{j}\oplus {a}_{k}\right)\right]$ $\underset{_}{\otimes }\equiv \left[\otimes ,\oplus \right]$ $↔$ composed operation 6) If $\left\{\left[A=\left(a\oplus b\right)\right],\left[B=\left(c\oplus d\right)\right]\right\}$ $\to$ $\left(A\underset{_}{\otimes }B\right)=\left(a\oplus b\right)\otimes \left(c\oplus d\right)$ 7) $\left(a\oplus b\right)\otimes \left(c\oplus d\right)=\left(a\otimes c\right)\oplus \left(a\otimes d\right)\oplus \left(b\otimes c\right)\oplus \left(b\otimes d\right)$ 8) $\left[\left({a}_{i}\otimes {a}_{j}\right)\oplus {a}_{k}\right]\ne \left[\left({a}_{j}\otimes {a}_{i}\right)\oplus {a}_{k}\right]↔{\left({a}_{i}\otimes {a}_{j}\right)}_{\oplus }\ne {\left({a}_{j}\otimes {a}_{i}\right)}_{\oplus }$