What is the meanning of this propisition ?

[Maths Lover]

hi , this proposition is from artin's book ,

it says :
Proposition : suppose an associative law of composition is given on a set S.
There is a unique way to define , for every integer n , a product of n elements
a1, …… , an of S ( we denote it by [a1 …an ] ) with the following properties :
-The product [a1] of one element is the element itself
-The product [a1 a2] of two elements is given by the law of composition ;
-For any integer I between 1 and n , [a1 .. an]=[a1….ai][a(i+1)…..an]


and I didn't understand excatly the meaning which the proposition says , so can anyone make it clear to me please ? and if there an example it'll be more better :)

 

[AlephZero]

Here's a simple example, using addition of real numbers.

You only define what addition means for adding two numbers (not more than two).

The associative law says that (x + y) + z = x + (y + z) for any three numbers x, y, and z.

The proposition says that if the associative law is true, there is a unique "answer" for the sum of ANY number of integers, a1 + a2 + a3 + ... + an, and the answer does not depend how you split the sum up into pairs of numbers using ()'s.

 

[Maths Lover]

Here's a simple example, using addition of real numbers.

You only define what addition means for adding two numbers (not more than two).

The associative law says that (x + y) + z = x + (y + z) for any three numbers x, y, and z.

The proposition says that if the associative law is true, there is a unique "answer" for the sum of ANY number of integers, a1 + a2 + a3 + ... + an, and the answer does not depend how you split the sum up into pairs of numbers using ()'s.

thank you very much :)
I have understood :)