A vector space basis is a set of vectors that span the space and is linearly independent.
It is well-known that for finite dimensional vector spaces this is equivalent to:
- The set is minimal with respect to being a spanning set.
- The set is maximal with respect to being linearly independent.
- Every element of the vector space is uniquely represented as a linear combination of the basis' vectors.
Now consider finite trees. A tree is defined as a set of edges (for some fixed vertex set) such that they connect the vertices (e.g. the tree is a connected graph) and the set is "independent" in the sense that there are no cycles.
Again, this is equivalent to:
- The set is minimal with respect to the connectedness of the graph.
- The set is maximal with respect to the independence (any edge added would create a cycle).
- For every two vertices, there is a unique path between them in the tree.
The similarity is very striking, I believe.
My question is - is this simply a coincidence or part of a greater (category-theory?) connection I'm unaware of? And are there more instances in mathematics for sets having this "minimal-maximal-unique" set of defining properties?