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In my context, I encounter a lot of partial orders with the distinguished property that the order is total on connected components. Equivalently, they satisfy the condition $$x \le y,z \enspace \lor \enspace y,z \le x \qquad \Longrightarrow \qquad y \le z \enspace \lor \enspace z \le y.$$

Is there an established term for partial orders with this property?

Context: I'm looking for a single adjective that will allow me to say "[adjective] ordered group" by analogy with "totally ordered group". My current choice is to say "fully ordered group" in the draft, but I'd like to switch to the established term if there is one.

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    $\begingroup$ I think you need to add the dual condition too it to be equivalent. $\endgroup$
    – Sam Hopkins
    Commented Jan 19, 2020 at 19:55
  • $\begingroup$ @SamHopkins: of course, thanks, I'll make the edit. $\endgroup$
    – Tobias Fritz
    Commented Jan 19, 2020 at 19:57
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    $\begingroup$ Anyways, "disjoint union of total orders" is concise enough that I doubt there is another name for these. $\endgroup$
    – Sam Hopkins
    Commented Jan 19, 2020 at 19:59
  • $\begingroup$ @SamHopkins: I'm looking for a single adjective that would capture this property, so that I can say "(whatever) order" in a similarly convenient way as I can say "total order". More concretely, I want to be able to say "(whatever) ordered group", just as I can say "totally ordered group". Saying "partially ordered group which is a union of total orders" sounds quite clumsy. I'm currently going with "full order" in the draft, but I'd like to change it to the established term if there is one. $\endgroup$
    – Tobias Fritz
    Commented Jan 19, 2020 at 20:06
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    $\begingroup$ I believe Sam Hopkins is right, but if you want a more concise adjective, how about “component-wise totally ordered group”? $\endgroup$
    – Emil Jeřábek
    Commented Jan 19, 2020 at 21:24

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