ATRACTOR

On the set of positive integers we have a well known order \[1<2<3<4<5<...\]

But there are other ways of comparing two positive integers: we will write \(d\trianglelefteq a\) (or \(a\trianglerighteq d\)) when \(d\) is a divisor of \(a\) (which is the same as saying that \(a\) is a multiple of \(d\), which means that there exists \(b\) such that \(bd=a\)).

It is clear that if \(d\trianglelefteq a\) then \(d\leq a\) (every positive integer is greater or equal than any of its divisors).

The relation \(\trianglelefteq\) has an important property in common with the usual order:

if \(a\trianglelefteq b\) and \(b\trianglelefteq c\) then \(a\trianglelefteq c\).

But there is another important property of the usual order that \(\trianglelefteq \) doesn't have: for any two positive integers \(a\) and \(b\) one has \(a\leq b\) or \(b\leq a\); however, one doesn't have, for any \(a\) and \(b\), \(a\trianglelefteq b\) or \(b\trianglelefteq a\) (for example, one has neither \(2\trianglelefteq 3\) nor \(3\trianglelefteq 2\)).

We can use the same language for \(\trianglelefteq \) as we use for the usual order:

• \(4\) is bigger than \(2\) for \(\trianglelefteq \)

• \(3\) is not bigger than \(2\) for \(\trianglelefteq \) (although \(3\) is bigger than \(2\) for the usual order)

• \(6\) is smaller than \(12\) for \(\trianglelefteq \)

• the maximum of the set \(\left\{ 2,4,6,12\right\} \) for \(\trianglelefteq \) is \(12\) (because \(12\trianglerighteq 2\) and \(12\trianglerighteq 4\) and \(12\trianglerighteq 6\) and \(12\trianglerighteq 12\))

• the minimum of the set \(\left\{ 2,4,6,12\right\} \) for \(\trianglelefteq \) is 2 (because \(2\trianglelefteq 2\) and \(2\trianglelefteq 4\) and \(2\trianglelefteq 6\) and \(2\trianglelefteq 12\))

• the set \(\left\{ 2,3,6\right\} \) has a maximum for \(\trianglelefteq \) (\(6\)) but it doesn't have a minimum for \(\trianglelefteq \).

It is easy to see that, if a set has a maximum for \(\trianglelefteq \), this maximum is a maximum for the usual order, too. The converse may not be true.

Generally we use a geometrical representation for the usual order - we represent the positive integers in a line, with the smaller on the left and the bigger on the right: \[1-2-3-4-5-6-7-...\]

In the same way we can represent the integers for the order \(\trianglelefteq \), again with the smaller (for \(\trianglelefteq \)) on the left and the bigger (for \(\trianglelefteq \)) on the right. The difference is that now they can't be represented in a line (which has to do with the fact that we don't always have \(a\trianglelefteq b\) or \(b\trianglelefteq a\)). Furthermore, the scheme quickly becomes very overloaded. We can therefore see this representation only for some finite sets; with this representation it is easy to see if a set has a maximum for \(\trianglelefteq \).