Does associativity of binary operation imply closure under this operation?
Sometimes definitions of semigroup, group or vector space omit axiom of closure under corresponding operations and sometimes they don't.
One of the arguments for omitting the axiom that I found is that associativity implies closure.
As a possible proof, let + be binary operation on set A. Assume a, b and c are elements of A. Also (b + c) is in set but (a + b) is not in a set.
Then a + (b + c) is well-defined (even though the result can be out of set).
However, if we assume that + is associative, we will get:
a + (b + c) = (a + b) + c
But that is not true because second "addition" in right part is not defined for (a + b) that is out of set and c.
So for associativity, (a + b) must be in set.
Does this argument make sense? Is it true?
UPDATE: In a possible proof the error is in the first assumption. If + is binary operation on A and a and b are in A then (a + b) must be in set by definition of binary operation (A x A -> A).