What is directed union and it holds algebraic structure ?
My understanding may be incorrect, I would be appreciated if you could correct my understandings.
My understanding; Directed union of set $H_i$($i∈I$) is defined as union of $H_i$ where for all $i,j∈I$,$i≧j$ implies $H_i≧H_j$.
For example, union $ \bigcup_{n≧1} \Bbb{F}_q$ is directed because $n｜m$ implies $\Bbb{F}_{p^n}⊆\Bbb{F}_{p^m}$.
Directed union of fields are again a field, because operation result is in for some large $n$, in $H_n$.
In this context, $A \geq B$ is frequently read as "A is farther than B."
A directed set is a set equipped with a relation $\geq$ that satifies:
i.) A is farther than A for all A $\\$
ii.) If A is farther than B and B is farther than C than A is farther than C $\\$
iii.) Given any two elements A and B there exists an element C that is further than both of them. That is, C is further than A and C is further than B.
A function defined on a directed set is known as a "net". Nets generalize the concept of sequences, and in turn allow the concept of limits to be generalized. (e.g. The Riemann integral is actually a limit of a net of Riemann Sums. When you define the integral this way many of the theorems on integrals become trivial to prove. For example, that the integral of a sum is equal to the sum of the integrals is an automatic consequence of the fact that the limit of a sum is equal to the sum of the limits). So it is more the topological structure of directed sets that tends to be useful than it is anything algebraic.
A directed union is a directed set that inherits its ordering from the index set in the manner you have described, although you should write it as $i \geq j$ implies $H_i \supseteq H_j$. Property iii.) above guarantees that the union will be closed with respect to the algebraic operation in question (in particular when you are adding/multiplying elements from different sets in the union).