Well, this depends on the definition of the sgn function. I would say

should be

, so

.

However, I don't see how this changes anything. If, for some function

and some transitive relation

, the relation

is defined as

, then

is transitive. If we have some unusual case, e.g., when

is not defined, then we cannot say

for any

. If

and

are defined but

is not, then again

and

are not related, so the premise of the transitivity property is false. What is important is for

to be a function, i.e., return exactly one value for each

.

Do you have any counterexample in mind?