The condition for associative property

Question

Thank in advance.

Is there the condition for associative property in closed number system. It comes from a question. A number system is closed, associative, commutative for some operation, then the system is closed for modulus operation.

For my test, I make some number system, i.e., finite, closed, commutative. But I don't know how to make a number system satisfying associative property.

//////////////// edited /////////////////

Sorry for the confusion.

Given a finite abelian group with binary operation *, let a*b = A and c*d = B.

If a (mod 2^16) = c (mod 2^16) and b (mod 2^16) = d (mod 2^16),

then A (mod 2^16) = B (mod 2^16).

This proposition is true? It's my original question.

Answer 1

Your number system is a group with some binary operation (*) on it.

Now remember that a set S with * is a group $\langle S, * \rangle$ if and only if:

• The operation * is well-defined and closed on S.
• There exists an identity $e \in S$.
• For every $a \in S$, there exists an $a^{-1}$ such that $a * a^{-1} = a^{-1} * a = e$.
• For all $a,b,c \in S$, we have $a*(b*c) = (a*b)*c$.

Associativity is actually one of the conditions a set and operation must fulfil in order to be a group --- so associativity holds for your "number system" by definition.