Most of the point of this answer is to promote a piece of terminology:

Three years ago I first taught a number theory course at UGA in which I made the following definition: a subset $A$ of the positive integers is substantial if

$\sum_{n \in A} \frac{1}{n} = \infty$.

A little bit of discussion of this concept occurs in Section 4 of

http://math.uga.edu/~pete/4400primes.pdf

I do mention the Erdos(-Turan?) conjecture about arithmetic progressions in substantial sets. For the purpose of constructing examples, possibly the remark I make about any set with positive upper density being substantial will be of most use to you.

Those notes don't contain a proof of that, but a proof of this and more can be found in a very (very!) nice final project done in this class by (then) undergraduate Alex Rice. Sadly he never gave me an electronic copy in a form that I was able to upload to my webpage. If you want to see his writeup, let me know and I'll bug him about this again: the winds of fate have blown him around a bit, but he is now again a UGA (graduate) student taking a number theory course from me.

Finally, to answer one of your questions in a cheap way: yes, the set of substantial subsets of $\mathbb{Z}^+$ certainly forms a semigroup under union. This seems like a completely unhelpful observation, but who knows...

Most of the point of this answer is to promote a piece of terminology:

Three years ago I first taught a number theory course at UGA in which I made the following definition: a subset $A$ of the positive integers is substantial if

$\sum_{n \in A} \frac{1}{n} = \infty$.

A little bit of discussion of this concept occurs in Section 4 of

http://alpha.math.uga.edu/~pete/4400primes.pdf

I do mention the Erdos(-Turan?) conjecture about arithmetic progressions in substantial sets. For the purpose of constructing examples, possibly the remark I make about any set with positive upper density being substantial will be of most use to you.

Those notes don't contain a proof of that, but a proof of this and more can be found in a very (very!) nice final project done in this class by (then) undergraduate Alex Rice. Sadly he never gave me an electronic copy in a form that I was able to upload to my webpage. If you want to see his writeup, let me know and I'll bug him about this again: the winds of fate have blown him around a bit, but he is now again a UGA (graduate) student taking a number theory course from me.

Finally, to answer one of your questions in a cheap way: yes, the set of substantial subsets of $\mathbb{Z}^+$ certainly forms a semigroup under union. This seems like a completely unhelpful observation, but who knows...