A lattice (L, ≤) is a partially ordered set, where each two elements of the set have a least upper bound and a greatest lower bound. Let's consider the situation where L is finite.
I think that this implies that each finite lattice has single greatest upper bound and a single least lower bound; the lower one can be defined as follows:
∃ min(L) ∊ L: ∀ x ∊ L-min(L) : min(L) < x
Is this true? If it's true; I have been trying to prove it; starting from two points:
1. Such min element must exist.
2. If it exists, it must be unique.
Any ideas? Or is there any reference that states this conclusion explicitly?
HINT: Show by induction that if $L$ is any lattice, and $F\subseteq L$ is finite, then $\bigwedge F$ and $\bigvee F$ exist in $L$. Conclude immediately that if $L$ is finite, it has a minimum element, $\bigwedge L$, and a maximum element, $\bigvee L$. Uniqueness is clear: if $x$ and $y$ are both greatest lower bounds (least upper bounds, resp.) of $L$, then $x\le y$ and $y\le x$, so $x=y$.