To prove that every non-empty finite lattice $L$ is complete, we need to show that every subset of $L$ has both a least upper bound (supremum) and a greatest lower bound (infimum).

Proof

1. Existence of Bounds: Let $L$ be a non-empty finite lattice. By definition, a lattice means that for any two elements $a, b \in L$, both $a \vee b$ (supremum) and $a \wedge b$ (infimum) exist.

2. Subset Consideration: Consider any subset $S \subseteq L$. Since $L$ is finite, $S$ can also be finite. We need to show that $S$ has a supremum and infimum in $L$.

3. Order Relation: The set $S$ can be ordered based on the relation defined in the lattice. Let $\mathcal{M}$ be the set of all upper bounds of $S$ in $L$. Since $L$ is finite, $\mathcal{M}$ is also finite.

4. Finding Supremum: The set $\mathcal{M}$ has a minimum element $m$ because any finite set of elements in a totally ordered set has a least element. This $m$ serves as the least upper bound of $S$ (i.e., $m = \sup S$).

5. Lower Bounds: Similarly, let $\mathcal{N}$ be the set of all lower bounds of $S$. Again, $\mathcal{N}$ is finite, and by the same reasoning, it has a maximum element $n$, which serves as the greatest lower bound of $S$ (i.e., $n = \inf S$).

6. Conclusion: Since $S$ was arbitrary, we conclude that every non-empty finite subset of $L$ has both a supremum and an infimum. Thus, every non-empty finite lattice $L$ is complete.

Final Result

Therefore, we have proven that every non-empty finite lattice is complete.

Would you like further details on any specific step or concept? Here are some related questions:

1. What are the implications of a lattice being complete?
2. How does the concept of completeness apply to infinite lattices?
3. Can you provide examples of complete lattices?
4. What is the significance of the least and greatest elements in a lattice?
5. How do finite lattices differ from infinite lattices in terms of structure?

Tip: Visualizing lattices with Hasse diagrams can greatly aid in understanding their structure and relationships.