Flat lattice

In mathematics, in the area of order theory, the flat lattice on a multielement set is the smallest lattice containing the elements of in which those elements are all pairwise incomparable; equivalently, it is the rank-3 lattice where is exactly the set of elements of intermediate rank.

Formal definition

As a partially ordered set, the flat lattice on a multielement set is given by where
As an algebraic structure, the flat lattice is equivalently given by where
and symmetrically for join:
All of the cases in the algebraic definition follow from the general lattice axioms except for the cases; the poset definition implies them because elements of are incomparable, so is the only remaining possibility for the value of the meet.
Similarly, the algebraic definition implies the poset definition because no two distinct elements have a nontrivial meet or join, so the only relations that are possible are those that are mandatory from the lattice axioms—exactly the three disjuncts listed.
Finally, the flat lattice on may be defined implicitly, as the least lattice in which the elements of are incomparable. This too is equivalent: Consider any lattice on where the set's elements are pairwise incomparable. Because they are multiple and incomparable, none of them can be the lattice top or bottom, and since is nonempty, and must be distinct. Therefore, is the smallest possible set such a lattice could be defined on, and incomparability and the lattice axioms wholly determine the element relations to match the definitions above.

The flat lattice of an empty set or of a singleton

The three definitions above, however, do not agree when is the empty set or a singleton, as the implicit definition would then collapse lattice elements. Possible resolutions are to exclude these cases, to define the flat lattice of such a set to be the one-element lattice or to define it as the two- or three-element lattice.