Say I have a relation that is Irreflexive and Transitive, but neither Symmetric nor Anti-Symmetric, can it still be a strict partial and / or strict total ordering? I realise this is an edge case, I appreciate any comments :)

  • $\begingroup$ Well, if we take a non-strict order and remove all relations $x\leqslant x$, we get a strict order. If we remove some of those and leave some others, we end up with your case. $\endgroup$ – Ivan Neretin May 21 at 7:14
  • $\begingroup$ Thank you very much! I couldn't get a definitive answer through googling. Much appreciated! $\endgroup$ – myName May 21 at 7:18
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    $\begingroup$ If it's irreflexive and transitive then it has to be antisymmetric. $\endgroup$ – Asaf Karagila May 21 at 7:51
  • $\begingroup$ It's also worth learning about partial orders, for which some $x\ne y$ satisfy neither $x\lt y$ nor $y\lt x$. $\endgroup$ – J.G. May 21 at 7:52

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