3
$\begingroup$

Can $\gcd(a,b)$ be called a binary operator which takes operands $a$ and $b$ and returns their greatest common divisor.

And if for some operator —say $\bigotimes$$(a_1\bigotimes a_2 \bigotimes ... ... \bigotimes a_{n-1} \bigotimes a_n)$ $= (a_1\bigotimes a_2...\bigotimes a_m)\bigotimes (a_{m+1}\bigotimes a_{m+2}...\bigotimes a_n)$, what is this property called?

  • 1
    Yes. The first expression isn't well-defined for a binary operator unless it's associative. If it is, then the property expresses a special case of associativity.2011-12-12
  • 3
    Yes, you can call it a binary operation (if you are working in a setting where any pair of elements have a "greatest common divisor" that is somehow uniquely defined). As to your second query, that would be "associativity".2011-12-12
  • 0
    @Qiaochu I might call that expression uniquely defined if there is a left-to-right reading convention. I think that might be the OP's intent, given the question.2011-12-12
  • 0
    I think that this is not as much about notation, but rather terminology and definition.2011-12-12
  • 4
    Indeed, it is a binary operation. The positive integers are something called a lattice under the operation of divisibility, and in lattice theory, we usually write the greatest common lower bound as $\wedge$, as in $a\wedge b$. In this lattice, $a\vee b$ is the least common upper bound, and is the least common multiple.2011-12-12
  • 0
    @ThomasAndrews Is lattice an abstract mathematical construct like group,ring etc.? Does grasp of these abstract structures help in other areas of maths?2011-12-12
  • 2
    Yes, a lattice is an abstract concept like group. It's hard to say what you mean by "help in other areas of math," but lattices come up a fair amount in lots of areas of math, including ring and group theory.2011-12-12

0 Answers 0