Flat lattice

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 ${\displaystyle X}$ is the smallest lattice containing the elements of ${\displaystyle X}$ in which those elements are all pairwise incomparable; equivalently, it is the rank-3 lattice where ${\displaystyle X}$ is exactly the set of elements of intermediate rank.^[1][2][3]

Formal definition

As a partially ordered set, the flat lattice on a multielement set ${\displaystyle X}$ is given by^[2] ${\displaystyle (X\cup \{\top ,\bot \},\preceq )}$ where $${\displaystyle \forall a,b.a\preceq b\leftrightarrow a=\bot \lor a=b\lor b=\top .}$$

As an algebraic structure, the flat lattice is equivalently given by^[3] ${\displaystyle (X\cup \{\top ,\bot \},\sqcap ,\sqcup )}$ where $${\displaystyle \forall a,b.a\sqcap b={\begin{cases}\bot &{\text{if }}a=\bot \lor b=\bot \\\bot &{\text{if }}a\in X\land b\in X\land a\neq b\\a&{\text{if }}a=b\\a&{\text{if }}b=\top \\b&{\text{if }}a=\top \\\end{cases}}}$$ and symmetrically for join: $${\displaystyle \forall a,b.a\sqcup b={\begin{cases}\top &{\text{if }}a=\top \lor b=\top \\\top &{\text{if }}a\in X\land b\in X\land a\neq b\\a&{\text{if }}a=b\\a&{\text{if }}b=\bot \\b&{\text{if }}a=\bot \\\end{cases}}}$$

All of the cases in the algebraic definition follow from the general lattice axioms except for the ${\displaystyle a\in X\land b\in X\land a\neq b}$ cases; the poset definition implies them because elements of ${\displaystyle X}$ are incomparable, so ${\displaystyle \bot }$ (respectively ${\displaystyle \top }$) 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 ${\displaystyle X}$ may be defined implicitly, as the least lattice in which the elements of ${\displaystyle X}$ are incomparable. This too is equivalent: Consider any lattice on ${\displaystyle X}$ 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 ${\displaystyle X}$ is nonempty, ${\displaystyle \top }$ and ${\displaystyle \bot }$ must be distinct. Therefore, ${\displaystyle X\cup \{\top ,\bot \}}$ 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 ${\displaystyle X}$ 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).