Difference between total order and linear order I want to ask a clarificatory doubt. I am really confused with the two definitions of linear order. In one definition it is said to be same as total order but in the other definition it is irreflexive part of total order (which also named as strict total order).
So my question is whether by definition linear order is reflexive (in that case is equivalent to total order) Or irreflexive (in this case it is not equal to total order).
To put it differently which one is a linear order (i) Less than and equal to, or (ii) less than.
Thank you.
 A: Linear order or total order are the same thing.
The difference between being reflexive or irreflexive depends just on a detail about how you want to think about order relations (whether total or not).
In the case of irreflexive orders (either partial or total), these are called strict, as you can see in the linked page on a comment above.
You shouldn't worry too much about this, since these are always interchangeable:
if $\leq$ is a (reflexive) partial order on a set $X$, you can obtain its strict (irreflexive) version $<$ by defining $x<y$ iff $x \leq y$ and $x \neq y$;
conversely, if $<$ is a strict partial order (irreflexive), you can obtain the corresponding reflexive version by defining $x \leq y$ iff $x<y$ or $x=y$.
These procedures are mutual inverses, meaning that if you start with one version of the order and apply both (each one once), you end up with the same order you started with (this is rather trivial to observe).
Notice that all this is valid for any partial order, whence, in particular, for a total order.
If some author has a different definition for total order and linear order, that can be OK, but I never saw that distinction.
One definition (reflexive or irreflexive) is not more correct than the other; it just happens that there are authors which prefer one over the other, and sometimes there are areas in which one is more widely used.
