Examples of coideals on $\omega$ I'm reading a paper that talks about selective coideals on $\omega$, but I can't seem to find much resources/notes on coideals. I understand that a coideal is a slight generalisation of an ultrafilter, but I can't wrap my head around what a coideal that is not an ultrafilter "looks" like.
Is it possible to give an explicit coideal that is not an ultrafilter?
EDIT: The paper I'm reading is Happy and mad families in $L(\mathbb{R})$. The definition of $H$ being a coideal, in page 2-3, is that $H$ satisfies the following two properties:

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*(Upward-closure) If $x \in H$ and $y \supseteq x$, then $y \in H$.


*(Pigeonhole) If $x_0 \cup \cdots \cup x_n \in H$, then $x_k \in H$ for some $k$.
 A: The reason for the terminology "coideal" is that $H\subseteq \mathcal{P}(X)$ is a coideal on $X$ if and only if its complement $I = \mathcal{P}(X)\setminus H$ is an ideal on $X$. [Well, except that an ideal should always contain $\varnothing$, so I think a coideal should additionally be assumed to not contain $\varnothing$.]
On the other hand, $F\subseteq \mathcal{P}(X)$ is a filter on $X$ if and only if its set of complements $I = \{X\setminus A\mid  A\in F\}$ is an ideal on $X$. If $U$ is an ultrafilter, then $U$ is both a filter and a coideal, since $A\notin U$ if and only if $X\setminus A\in U$, so the complement of $U$ is equal to the set of complements of $U$.
In general, if you start with an ideal $I$, you get a coideal by taking the complement of $I$, and you get a filter by taking the set of complements of $I$. Intuitively, if an ideal specifies a notion of "small" set, and a filter specifies a notion of "large" set, then a coideal specifies a notion of "not small" set.
So to find examples of coideals which are not ultrafilters, you just need to find examples of ideals which are not maximal ideals, and take their complements. For example:

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*The improper coideal $\varnothing$.

*The trivial coideal $\{A\subseteq X\mid A\neq \varnothing\}$

*The coideal generated by $A\subseteq X$: $\{B\subseteq X\mid B\not\subseteq A\}$.

*The coideal of infinite subsets of $X$.

*On $\mathbb{R}$, the coideal of non-null sets and the coideal of non-meager sets.

*On $\omega$, the coideal of non-summable sets $\{A\subseteq \omega\mid \sum_{n\in A}\frac{1}{n+1} = \infty\}$.

*On $\omega$, the coideal of non-zero-density sets $\{A\subseteq \omega\mid \lim_{n\to\infty} \frac{|A(n)|}{n}\neq 0\}$, where $A(n) = A\cap \{0,\dots,n-1\}$. We include the case when the limit does not exist.

