Does connexity imply reflexivity?

Question

The Wikipedia article on Total Order states, "The connex property also implies reflexivity, i.e., a ≤ a" without giving an explanation (and confusingly uses the "less than or equal to" symbol instead of R). However, there seems to be valid counterexamples; for instance, "less than" or "greater than" would appear to be connex but not reflexive. Is Wikipedia then incorrect, or am I misunderstanding?

Answer 1

You are misunderstanding: the usual $<$ relation on $\Bbb Z$ is not connex. In order for it to be connex, it would have to be true that for all $a,b\in\Bbb Z$, $a<b$ or $b<a$, but if we take $a=b=0$, for instance, neither of these is true.

More generally, suppose that $\preceq$ is a linear order on $A$, and let $a\in A$. $\preceq$ is connex, so either $a\preceq a$, or $a\preceq a$. In other words, $a\preceq a$, and $\preceq$ is therefore reflexive.