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It is well known that a directed set is a set equipped with a partial order such that every pair of elements admits an upper bound. My question is the following: when imposing conditions such as the last one on pairs of elements in technical mathematical language, does one mean that the pair must be composed of distinct elements of does one refer to the elements of the cartesian product of the set with itself, thereby including the pairs made of the same element? Thank you for your attention.

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    $\begingroup$ When stating such a condition, one may require the two to be different, or not; but if you don’t explicitly say the two must be different, then it is understood that you allow them to be the same. You are saying “For all $x$ and for all $y$, $P(x,y)$”, and that does not preclude $x=y$. If you want to preclude the case $x=y$, then you need to say so explicitly. $\endgroup$ – Arturo Magidin Mar 22 at 0:08
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As Arturo already explained in a comment: unless explicitly stated otherwise, we do include pairs of consisting of two times the same element.

For the definition of directed set however, this does not matter. In a preorder $(P, \leq)$, any pair $(a, a)$ has a common upper bound, namely $a$ itself. So the pairs consisting of two times the same element automatically satisfy this condition.

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