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Is the unique lattice on the empty set, also a complete lattice? More precisely, $(\emptyset, \emptyset)$ is the empty ordering, which is also a lattice. Is it also a complete lattice?

user107952
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    No. Complete lattices need top and bottom elements. (I also wouldn't call the empty poset a lattice; for me lattices should also have top and bottom elements.) – Qiaochu Yuan Sep 18 '17 at 23:59
  • @QiaochuYuan: if your lattices have to have tops and bottoms, then your lattices are other people's bounded lattices. – Rob Arthan Sep 19 '17 at 00:19
  • @NoahSchweber Apologies for my careless reading. – bof Sep 19 '17 at 02:16

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While the empty lattice is indeed a lattice - vacuously: any two elements have least upper bounds and greatest lower bounds and these behave the way they should - it is not, however, complete: the emptyset has no least upper bound, or greatest lower bound.

Note that complete lattices are bounded: the least upper bound of the emptyset has to be the minimal element, and the greatest lower bound of the emptyset has to be the maximal element.

Noah Schweber
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  • Whether the empty set is a lattice depends, I guess, on whether you consider a lattice to be a kind of ordered set or a kind of algebraic structure. – bof Sep 19 '17 at 02:15
  • @bof How do you mean? In some contexts, we allow algebraic structures to be empty (and it sounds like the OP is working in such a context). – Noah Schweber Sep 19 '17 at 02:24
  • You learn something new every day. I learned two things today: that the universe of an algebraic structure can be empty, and that "emptyset" is just one word. Sadly, I probably forgot more than two things today that I used to know, and I don't know what they are. – bof Sep 19 '17 at 07:40
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    @bof: "empty set" is two words. And whether or not an algebra can have an empty carrier set is a moot point. I think most writers on universal algebra disallow this, e.g. See https://www.math.uwaterloo.ca/~snburris/htdocs/UALG/univ-algebra2012.pdf – Rob Arthan Sep 20 '17 at 00:08
  • @RobArthan Thanks for the reference! – bof Sep 20 '17 at 00:36
  • It was a surprise for me to learn that the empty set can equip with a group structure. – Bumblebee Nov 17 '20 at 03:05
  • @Bumblebee That's not a group really - as they say, that's a generalization of the usual notion of a group. Because of the requirement that there be an identity element, groups cannot be empty. – Noah Schweber Nov 17 '20 at 03:21