Flat lattice

Short description: Smallest lattice in which the elements of a set are incomparable

Not to be confused with a lattice on the flats of a matroid, which is sometimes also called a "flat lattice".

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

Formal definition

As a partially ordered set, the flat lattice on a multielement set $X$ is given by $(X\cup \{\mathrm{\top },\mathrm{\perp }\},⪯)$ where $$\mathrm{\forall }a,b.a⪯b\leftrightarrow a=\mathrm{\perp }\vee a=b\vee b=\mathrm{\top }.$$

As an algebraic structure, the flat lattice is equivalently given by $(X\cup \{\mathrm{\top },\mathrm{\perp }\},\sqcap ,\bigsqcup )$ where $$\mathrm{\forall }a,b.a\sqcap b=\{\begin{array}{ll}\mathrm{\perp }& \text{if }a=\mathrm{\perp }\vee b=\mathrm{\perp }\\ \mathrm{\perp }& \text{if }a\in X\wedge b\in X\wedge a\ne b\\ a& \text{if }a=b\\ a& \text{if }b=\mathrm{\top }\\ b& \text{if }a=\mathrm{\top }\\ \end{array}$$ and symmetrically for join: $$\mathrm{\forall }a,b.a\bigsqcup b=\{\begin{array}{ll}\mathrm{\top }& \text{if }a=\mathrm{\top }\vee b=\mathrm{\top }\\ \mathrm{\top }& \text{if }a\in X\wedge b\in X\wedge a\ne b\\ a& \text{if }a=b\\ a& \text{if }b=\mathrm{\perp }\\ b& \text{if }a=\mathrm{\perp }\\ \end{array}$$

All of the cases in the algebraic definition follow from the general lattice axioms except for the $a\in X\wedge b\in X\wedge a\ne b$ cases; the poset definition implies them because elements of $X$ are incomparable, so $\mathrm{\perp }$ (respectively $\mathrm{\top }$) is the only remaining possibility for the value of the meet (respectively join).

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 $X$ may be defined implicitly, as the least lattice in which the elements of $X$ are incomparable. This too is equivalent: Consider any lattice on $X$ 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 $X$ is nonempty, $\mathrm{\top }$ and $\mathrm{\perp }$ must be distinct. Therefore, $X\cup \{\mathrm{\top },\mathrm{\perp }\}$ 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 $X$ is the empty set or a singleton, as the implicit definition would then collapse lattice elements. Possible resolutions are to exclude these cases (as above), to define the flat lattice of such a set to be the one-element lattice (in agreement with the implicit definition) or to define it as the two- or three-element lattice (in agreement with the explicit constructions).