Who knowsIf I define algebra F ( A ) generated by A, collection of subsets...

Who knows
If I define algebra $\mathcal{F}(A)$ generated by $A$, collection of subsets of $S$ (the universal set) as the intersection of $\mathcal{F}$, algebra superset of $A$:
$\mathcal{F}(A)=\bigcap _{algebra\mathcal{F}\supseteq A}\mathcal{F}$
What if $A$ is an infinite (either countable or uncountable) set? Algebra, unlike $\sigma$-algebra, guarantees being closed under finite Boolean operations. Here, finite(in the definition of algebra) and infinite(in the setting) confuses me. e.g. $A$ is the collection of intervals $(-\mathrm{\infty },x]$($x\in \mathbb{R}$) and $S=\mathbb{R}$, what $\mathcal{F}(A)$ be like? Any help will be appreciated!