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A lattice structure requires that every two elements have a join and a meet. Suppose we consider instead posets in which for every two elements $x,y$, if there exists an element greater than both of them, then their join exists, and if there exists an element less than both of them, then their meet exists. What can be said about such posets?

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  • $\begingroup$ I took my best guess as to what the question was asking. Of course Bjørn's answer shows that in some sense the question is trivial. $\endgroup$
    – Todd Trimble
    Commented Nov 11, 2014 at 17:23

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These structures can be characterized as being exactly what you obtain from bounded lattices by removing their top and bottom elements.

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