$T$ is defined on $\mathbb{N} \times \mathbb{N}$ by $(a,b)\mathrel{T}(c,d)$ if and only if $a \leq c$ and $b \leq d$.

I know this is a partial order relation as it is Transitive, Anti Symmetric and Reflexive but I'm not sure if its total order relation and whether they are well order relations?

  • $\begingroup$ try compare $(1,2)$ and $(2,1)$. $\endgroup$ – Ofir Schnabel Apr 29 '15 at 8:12

It is not a total order: just compare $(1,0)$ and $(0,1)$. It is a well partial order, a result known as Dickson's lemma.

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  • $\begingroup$ Im assuming a well order relation is different from a well partial order? As I don't think we have learnt Dickson's Lemma yet $\endgroup$ – Sunny Apr 29 '15 at 8:20
  • $\begingroup$ @Sunny. Look at the first sentence of this page on Well orders : In mathematics, a well-order relation (or well-ordering)... As a rule, please look at Wikipedia before asking any question on math.stackexchange. $\endgroup$ – J.-E. Pin Apr 29 '15 at 8:25

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