If all Total order relations are partial order as well, what about < relation?

Question

We know that < is total order as it satisfies all three properties antisymmetry, transitivity and connex but since it can not be reflexive we can say it will not be partial order.

But I have been reading that a total order is also a partial order

Answer 1

Here is the thing: $<$ is not a total order relation. It is a strict total order relation. These are irreflexive. Total orders are reflexive.

$<$ does not have the connex property. It is not true that $(a<b)\lor (b<a)$ for all $a$, $b$.

Rather, $<$ is trichotomous with equality: $(a< b)\oplus(b<a)\oplus(a=b)$.

Answer 2

Authors are a bit sloppy about what they mean by "total order", because it doesn't actually matter very much as long as context is clear.

Any weak total order (i.e. "reflexive antisymmetric transitive trichotomous relation") implies a strict total order by defining $x < y$ to be "$x \leq y$ and $x \not = y$".

Conversely, any strict total order implies a weak total order by defining $x \lesssim y$ to be "$x < y$ or $x = y$".