在意识到矩阵表示将导致物理量不满足乘法交换律之前，海森堡并没有前进太远。

Heisenberg had not proceeded very far with this idea before he noticed that it would lead to his physical quantities not satisfying the commutative law of multiplication.

我们容易验证乘法满足结合律和交换律，并且由加法结果的唯一性得出乘法结果的唯一性和乘法消去律。

We then can verify the associative law and the commutative law for multiplication, and the uniqueness of the result of addition indicates the uniqueness and the cancellation law for multiplication.

进一步的, 我们可以定义包含交换律 、 分配律 、 结合律等公理的代数.

Further on, we can specify an algebra that consists of axioms for commutativity, distributivity, associativity, etc.