:smack: Move on from their what? I can’t believe I wrote that. One day I’ll make a post that makes me look intelligent rather than illiterate.
Firstly, don’t worry, your questions aren’t stupid. These are difficult subjects. Just keep plugging away and keep asking questions when you get stuck and we’ll get there/their/they’re in the end.
Now let’s recap associativity. A product * is associative if, for every choice of a, b and c,
(ab)c = a(bc)
This means that to find the product of a, b and c ( in that order), we have two choices:
(i) Find the product ab. Call it d. The find the product dc.
(ii) Find the product bc. Call it e. Then find the product ae.
The associative law says that these two products will be equal.
To relate this to the earlier examples:
1/2/3) Addition and multiplication are associative.
4) Not associative: (a/b)/c is in general not equal to a/(b/c)
5)Not associative: (a - b) - c is not equal to a - ( b - c) in general.
A product * is called commutative if, for every a and b,
ab = ba
Associativity and commutativity are quite independent. In example 6, matrix multiplication is associative but not commutative. You cannot change the order of terms in a product unless * is commutative.
The operation of taking the composite of functions is associative. When we consider a set of functions it is almost always composition that we have in mind for the product. When using this as our product, we must take particular care to check that closure holds. So the force of the sentence you quoted in your OP is that the composition of two Lorentz transformations is again a Lorentz transformation. I would assume that this was proved just before the part you quoted, and that this is the “This” at the beginning of the quotation.
For the use of the word “a”, I would read the sentence as:
closed algebraic structure with ( a binary operation which is associative)
The “closed” is just the point I made in example 5 of my second post. The product of two elements of S again lies in S. The sentence is stressing that the product is associative.
Finally, you asked why sets of transformations are important. This comes from group theory. There is a result, called Cayley’s Theorem, which says that every group is isomorphic to a group of transformations ( that is 1-to-1 correspondences on a set).