When does it make sense to define a base of a set system? In a topology, a base is defined to be a class of subsets such that
    every open set is the union of some members of it. 
In a convexity
    structure, a base is defined to be a class of subsets such that
    every convex set is the union of an up-directed subcollection of it.
For a sigma algebra, an attempt to define a base is dismissed here. But I guess it may still be open for other possibilities to define what a base is.
A filter base generates a (proper) filter by including all sets which contain a set of the filter base.
So I was wondering whether a base can be defined for a general set
    system? When does a set system admit a base, and when doesn't?
Thanks and regards!
Some sources I have just found, although they haven't provided definitions:


*

*base for an algebra of sets: http://ncatlab.org/nlab/show/sigma-algebra#generating_algebras_14

*base for a sigma algebra of sets: http://ncatlab.org/nlab/show/base#bases_for_algebras_7
 A: I second William's comment that the concept of "base" means different things in different areas of math. So let me focus on the concept of "base" thought as "minimal set of elements that generate a structure".
There is a nice and simple categorical approach to this concept. Categorically, to give an algebraic structure, is to give a monad on a category $T \colon \mathcal{C} \rightarrow \mathcal{C}$. An algebra for a monad is an object in the category $\mathit{Alg}(T)$ of the Eilenberg-Moore resolution for the monad. This resolution comes equipped with a pair of adjoint functors: the forgetful functor $U \colon \mathit{Alg}(T) \rightarrow \mathcal{C}$ and its left adjoint "the free functor" $F \colon \mathcal{C} \rightarrow \mathit{Alg}(T)$ such that $T = U \circ F$.
Now, the crucial thing is that we may compose these functors in the other direction  obtaining a comonad $D = F \circ U$ on the category $\mathit{Alg}(T)$. As it is the case with every comonad, $D$ has its own coresolution in the category of the Eilenberg-Moore coalgebras $\mathit{coAlg}(D)$. Therefore, we may consider coalgebras over algebras. It turns out that in many cases such induced coalgebras over algebras correpond to the usual notion of base. For example: coalgebras over vector spaces decompose vectors into coefficients according to a base; coalgebras over algebras for the power-set monad give decomposition of elements of an atomic (complete) lattice into their descending atoms.

It may help to work out a simple example. Let us consider $\mathcal{C} = \mathbf{Set}$ together with a finite sequences monad:
$$T(X) = X^*$$
The Eilenberg-Moore resolution for $T$ gives the category of monoids $\mathbf{Mon}$. The forgetful functor $U \colon \mathbf{Mon} \rightarrow \mathbf{Set}$ assigns to a monoid its carrier:
$$U(\langle M, \bullet, \iota\rangle) = M$$
and the free functor $F \colon \mathbf{Set} \rightarrow \mathbf{Mon}$ gives a monoid action via concatenation:
$$F(X) = \langle X^*, \circ, [\;]\rangle$$
The induced comonad $D = F \circ U$ associates with every monoid the free monoid build upon the same carrier:
$$D(\langle M, \bullet, \iota\rangle) = \langle M^*, \circ, [\;]\rangle$$
There is the counit of the comonad $\epsilon \colon \langle M^*, \circ, [\;]\rangle \rightarrow \langle M, \bullet, \iota\rangle$ defined by folding with monoid's multiplication:
$$\epsilon([a_1, a_2, \dotsc, a_n]) = a_1 \bullet a_2 \bullet \cdots \bullet a_n$$
and the comultiplication of the comonad $\delta \colon \langle M^*, \circ, e\rangle \rightarrow \langle M^{**}, \circ, e\rangle$ which decomposes words on singletons:
$$\delta([a_1, a_2, \dotsc, a_n]) = [[a_1], [a_2], \dotsc, [a_n]]$$
By definition, a coalgebra over a monoid $\langle M, \bullet, \iota\rangle$ is a monoid homomorphism $h \colon \langle M, \bullet, \iota\rangle \rightarrow \langle M^*, \circ, [\;]\rangle$ such that:


*

*$\epsilon \circ h = \mathit{id}$, that is: $h(r)_1 \bullet h(r)_2 \bullet \cdots \bullet h(r)_n = r$, where $h(r)_k$ is the $k$-th element of sequence $h(r)$

*$D(h) \circ h = \delta \circ h$, that is: $h(h(r)_k) = [h(r)_k]$


These conditions say that coalgebras over monoids are tantamount to finite decompositions of elements on indecomposable elements.
