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$T$ is the set (say $S^\ast$) of all finite unions of finite intersections of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to finite unions. Finally, $S^\ast$ is closed wrt to finite intersections, because $$\bigcap_{j=1}^n\bigcup_{i_j\in I_j}A_{i_j,j} =\bigcup_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcap_{j=1}^n A_{i_j,j}$$ for any sets $A_{i,j}$ and any index sets $I_j$, and finite intersections of finite intersections of sets are finite intersections of sets.

Similarly/dually, $T$ is the set of all finite intersections of finite unions of elements of $S$.

Cf. conversion to the conjunctive normal form and the disjunctive normal form.

$T$ is the set (say $S^\ast$) of all intersections of finite unions of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to intersections. Finally, $S^\ast$ is closed wrt to finite unions, because $$\bigcup_{j=1}^n\bigcap_{i_j\in I_j}A_{i_j,j} =\bigcap_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcup_{j=1}^n A_{i_j,j}$$ for any sets $A_{i,j}$ and any index sets $I_j$ -- use now the displayed identity for the case when each of the $A_{i,j}$'s is the union of finitely many elements of $S$, and the fact that finite unions of finite unions of sets are finite unions of sets.

Cf. conversion to the conjunctive normal form and the disjunctive normal form.

$T$ is the set (say $S^\ast$) of all finite unions of finite intersections of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to finite unions. Finally, $S^\ast$ is closed wrt to finite intersections, because $$\bigcap_{j=1}^n\bigcup_{i_j\in I_j}A_{i_j,j} =\bigcup_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcap_{j=1}^n A_{i_j,j}$$ for any sets $A_{i,j}$ and any index sets $I_j$, and finite intersections of finite intersections of sets are finite intersections of sets.

Similarly/dually, $T$ is the set of all finite intersections of finite unions of elements of $S$.

$T$ is the set (say $S^\ast$) of all finite unions of finite intersections of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to finite unions. Finally, $S^\ast$ is closed wrt to finite intersections, because $$\bigcap_{j=1}^n\bigcup_{i_j\in I_j}A_{i_j,j} =\bigcup_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcap_{j=1}^n A_{i_j,j}$$ for any sets $A_{i,j}$ and any index sets $I_j$, and finite intersections of finite intersections of sets are finite intersections of sets.

Similarly/dually, $T$ is the set of all finite intersections of finite unions of elements of $S$.

Cf. conversion to the conjunctive normal form and the disjunctive normal form.

$T$ is the set of all finite unions of finite intersections of elements of $S$.

$T$ is the set (say $S^\ast$) of all finite unions of finite intersections of elements of $S$.

Indeed, clearly $T\supseteq S^\ast$. It is also clear that $S^\ast$ is closed wrt to finite unions. Finally, $S^\ast$ is closed wrt to finite intersections, because $$\bigcap_{j=1}^n\bigcup_{i_j\in I_j}A_{i_j,j} =\bigcup_{(i_1,\dots,i_n)\in I_1\times\cdots\times I_n}\,\bigcap_{j=1}^n A_{i_j,j}$$ for any sets $A_{i,j}$ and any index sets $I_j$, and finite intersections of finite intersections of sets are finite intersections of sets.

Similarly/dually, $T$ is the set of all finite intersections of finite unions of elements of $S$.