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As an example of ordering relation that do not contain a comparative term ( such as " is greater than") in its definition, I can find

" set X is included in set Y".

But it seems difficult to me to find many other examples.

Can you think of many relations that are orderings but do not contain any apparent comparative term in their definition?

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    $\begingroup$ This is essentially the same question you asked about equaivalence relations here: math.stackexchange.com/questions/3204326/… . The answer is the same as @JeanMarie 's answer there: an order relation must have a comparison operator, even if you call it something else. $\endgroup$ – Ethan Bolker Apr 27 '19 at 12:29
  • $\begingroup$ @EthanBolker.I understand this, but that does not prevent certain relations to be less trivially equivalence/ordering relations than others, does it? ( My question is asked at the beginner's level). $\endgroup$ – user654868 Apr 27 '19 at 12:37
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    $\begingroup$ Beginner's level is just fine. We were all beginners once. $\endgroup$ – Ethan Bolker Apr 27 '19 at 12:52
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The relation on the integers given by "$a$ divides $b$" is an order relation that does not seem to be described by a comparison operator.

You might say the same about the relation on people defined by "ancestor of". Whether that counts depends on whether you think of "ancestor of" as a kind of comparison.

As discussed elsewhere in this question, there is always a comparison, which may be more or less explicit.

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  • $\begingroup$ More precisely, the natural numbers ordered by divisibility form a lattice. $\endgroup$ – user76284 Apr 27 '19 at 21:05

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