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Is there a name for a distributive lattice with only two (nonequal) elements, 1 and 0, corresponding to top and bottom, respectively? I was thinking of calling it a trivial lattice, but a trivial lattice should probably just have one element.

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  • $\begingroup$ I'd call it the trivial bounded lattice myself, since bounded lattices by definition have a top and bottom element. (The single element lattice would be degenerate in that case) $\endgroup$
    – Alan
    Jul 17, 2021 at 19:37
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    $\begingroup$ Some call it $\mathbf 2$, and in general, $\mathbf n$ to a $n$-element chain. $\endgroup$
    – amrsa
    Jul 17, 2021 at 19:40
  • $\begingroup$ The free lattice on 0 elements $\endgroup$ Jul 17, 2021 at 20:07

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For a name, I would use the two-element chain. Grätzer uses this at least nine times in his book Lattice Theory. Foundation (Birkhäuser/Springer 2011).

Common notations are $C_2$ (used by Grätzer) and $\mathbf{2}$ (used by Stanley in Enumerative Combinatorics, vol. 1).

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