I am really sorry for my mistake. It should be "imply Transitivity" instead of "imply Reflectivity". Thank you, everyone.
I am reading Khovanov's lecture notes for representation theory of finite groups, and get confused.
He defines total order this way.
A totally ordered set $T$ is a set together with a binary relation $≤$ which is reflexive and antisymmetric and such that, for all $a$ and $b$ in $T$, either $a ≤ b$ or $b ≤ a$ (this property is called totality). In this case, the binary relation is called a total order on the set $T$.
Then he remarked: One can check that antisymmetry and totality imply reflexivity. Thus, a totally ordered set is equivalent to a partially ordered set in which the binary relation is total.
I understand why that totality implies reflexivity, but I can not get why Reflexivity, Totaility, and Antisymmetry imply Transitivity?
Here the link of the note. The remark is on the bottom of page1. https://pdfs.semanticscholar.org/ff34/cedfbc36821d57def15b59d13f2926bf385e.pdf
The claim about transitivity is false. Consider $A=\{a,b,c\}$ and the relation $\{(a,b),(b,c),(c,a),(a,a),(b,b),(c,c)\}$.
This relation is reflexive on $A$, obviously. It is also total, since every two elements are comparable. It's even antisymmetric. But it is not transitive, since $(a,b)$ and $(b,c)$ should imply $(a,c)$ in which case $(c,a)$ would imply $a=c$.
But the notes make no such remark. The notes simply remark that totality (and antisymmetry) imply reflexivity. But this is an easy observation, since given $a=b$, either $a\leq b$ or $b\leq a$, which in both cases we get $a\leq a$. The remark then continues, a totally ordered set is a partially ordered set which is also total. The two parts of the remark are somewhat independent and I would daresay it is even [pedagogically] wrong to put them so close to each other, as it can be misleading to the unsuspecting reader.
However, having written notes before, I can also attest that sometimes you write notes that don't make full sense, because you made some observation to yourself, and didn't sit down to chase its details in completeness. This leads to many of these small remarks that can be either misleading at best, or completely mistaken at worse cases.