From: Marc van Dongen <dongen@cs.ucc.ie> Date: Sun, 16 Dec 2001 13:35:59 +0000
Marc van Dongen (dongen@cs.ucc.ie) wrote:
: An integer $a$ divides integer $b$ if there exists an integer : $c$ such that $a c= b$.
To make clear why $0$ (and not any other non-zero integer) is the gcd of $0$ and $0$ I should have added that for the integer case $g$ is called a greatest common divisor (gcd) of $a$ and $b$ if it satifies each of the following two properties:
1) $g$ divides both $a$ and $b$; 2) if $g'$ is a common divisor of $a$ and $b$ then $g'$ divides $g$.
In case it isn't clear already, these definitions make a lattice on the positive integers, with divides ~ leq, gcd ~ meet and lcm ~ join, using the report's definitions of gcd and lcm. (Choosing the positive result for gcd/lcm is equivalent to noting that divides is a partial preorder on the non-zero integers, and that the quotient identifies x and -x). The only thing that is lacking to make it a lattice on the non-negative integers, is that gcd 0 0 = 0 . All other cases involving zero (i.e., gcd 0 x = x for non-zero x, and lcm 0 x = 0 for all x) are consistent with 0 being the maximal element in the lattice, i.e., that all integers divide zero. Lars Mathiesen (U of Copenhagen CS Dep) <thorinn@diku.dk> (Humour NOT marked)