Is the $=$ in the preorder $\geq$ identity or equivalence in the preorder?

Question

Question

Is the $=$ in the preorder $\geq$ identity or equivalence in the preorder?

Background

By way of example, let $\geq$ be the preorder on the cyclic set $\Bbb Z/3\Bbb Z$ having the chain $0,1,2,1\ldots$

There are two possible ways to break down and interpret the relation $a\geq b$:

1. We can write it as $a>b$ OR $a=b$ with equals meaning the identity relation. So $1=1$ and $2=2$

2. We can write it as $a>b$ OR $a\sim b$ with similar meaning "equal in the preorder", i.e. $1\sim1$, $2\sim2$, AND $1\sim 2$.

Which does it mean?

Answer 1

The symbol "$=$" in mathematics should always be reserved to denote actual equality. That is, the two objects on both sides are identical. This applies to the situation of preorders as well, so in your example $1\le 2$ and $2\le 1$ but still $1\neq 2$.