Let $\mu$ be any infinite cardinal, and define a collection $N\subset[\mu]^\mu$ to be, maximal almost disjoint (MAD) over $\mu$, iff

- $\forall\{A,B\}\in[N]^2$ $( A\cap B \in [\mu]^{<\mu})$
- $\forall X\in[\mu]^\mu \exists A\in N$ $( X \cap A \in [\mu]^\mu)$

My questions are as follows: when $\mu$ is **singular**

- What is known about MAD families (or any other combinatorial structures, like, towers, SFIP families without pseudo-intersection, etc) over $\mu$?
- Are such families degenerate in the sense that an infinite family can have cardinality below $\mu$?
- Is there any connection between such constructs on $\mu$ and the corresponding constructs on $cf(\mu)$?

The main point I really want to know is this: Is it possible to add new subsets to an arbitrary singular cardinal without adding new subsets the the cardinals below it?

Side Request: I've been told there are forcing constructions which will add an order type $\omega$ cofinal sequence to a cardinal with cofinality $\omega$, can anyone point me in the correct direction with a book or article?