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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?

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    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

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