Let $P$ be a poset and let $x$ and $y$ be elements of $P$. I have occasionally seen books define a notion of "compatible" elements in a poset, so that $x$ and $y$ are compatible iff $\exists a \in P$ such that $a \le_P x$ and $a \le_P y$ (i.e. there is a common lower bound).
Often, such a notion is used to define other notions (e.g. strong antichain) within the book. I have a few questions regarding potential ambiguity:
- I presume the above notion is common within the study order theory. Is the word "compatible" itself common in order theory? Or is it archaic and only found in old textbooks?
- According to the Wikipedia entry on "strong antichain", there is no convention that "strong antichain" refers specifically to "strong downwards antichain". Is there a convention such that "compatiblility" refers to common lower bound, or is it an ambiguous term?
- Is there a specific way to distinguish this notion from its dual? That is, can we explicitly say "compatibility from above" or "compatibility from below" in order to distinguish the two?
I am curious about the general ambiguity of terms with respect to "above" and "below". I would also like to know if there are alternative methods I can use to remove the ambiguity.
I know the notion of compatibility (understood as the existence of a common upper bound) from Mark V. Lawson's book "Inverse Semigroups: The Theory of Partial Symmetries" (inverse semigroups being equipped with a natural order, order theory finds some applications in this structure). However I don't know any "standard" reference of "pure" order theory making use of this notion.