Math 2.0

Order, lattices, ordered algebraic structures Definition

Order on the integers

Builds on IntegersNatural numbers

Definition

Let mm and nn be integers. The notation m≤nm \le n means that n−mn - m is a natural number, that is, when n−m∈{0,1,2,… }n - m \in \{0, 1, 2, \dots\}. The notation m<nm < n means that m≤nm \le n and m≠nm \neq n.

Remarks

This relation is a total order on ℤ\mathbb{Z}, and on ℕ\mathbb{N} it agrees with the order on the natural numbers. If m≤nm \le n, then m+k≤n+km + k \le n + k for each integer kk. If m≤nm \le n and 0≤k0 \le k, then mk≤nkm k \le n k. Thus ℤ\mathbb{Z} is an ordered ring. Unlike ℕ\mathbb{N}, it has no least element.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Integer, Order-theoretic properties
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