About $p$-filters with $p\in\beta X\setminus X$. I was reading this article and the next underlined part took my atention

The definition of $p$-filter sounds so different from the usual definition of a filter in a topological space. I know that maybe this can be a different definition and maybe the only thing related with filters is the name. Is there another place where I can read about this "$p$-filters? Or is this only a definition concerning this article and nobody worked with that? I am really interesing about to know more  about $p$-filters. I really apreciated any help. Thanks.
 A: If I understand the definitions correctly, $\mathscr{F}$ is not necessarily a filter in the usual sense, but if you close it under finite intersections and then under supersets, you get an ordinary filter on $\pi$ that is still a $p$-filter on $\pi$.
Let $\mathscr{B}=\left\{\bigcap\mathscr{A}:\mathscr{A}\subseteq\mathscr{F}\text{ and }\mathscr{A}\text{ is finite}\right\}$; then $\mathscr{B}$ is closed under finite intersections, and $p\in\operatorname{Cl}\bigcup B$ for each $B\in\mathscr{B}$, so $\mathscr{B}$ is a $p$-filter on $\pi$ that contains $\mathscr{F}$. Finally, let $\mathscr{G}=\{G\subseteq\pi:B\subseteq G\text{ for some }B\in\mathscr{B}\}$; $\mathscr{G}$ is a filter on $\pi$ that is also a $p$-filter containing $\mathscr{F}$. (In fact it is the minimal such filter on $\pi$.)
Say that $U\subseteq X$ is a $\pi$-set if it is the union of members of $\pi$. Then $\left\{\bigcup G:G\in\mathscr{G}\right\}$ is a $\pi$-set filter on $X$ with the property that $p\in\operatorname{Cl}U$ for each $G\in\mathscr{G}$.
