Let $(L,\subseteq)$ be a complete lattice with elements denoted by the uppercase letters of the alphabet (suppose that each element stays in $\mathcal{P}(\Gamma)$ for some set $\Gamma$) and $\mathcal{A} \subseteq L$ be a subset of elements of the lattice, which relation exists between $inf(\mathcal{A})$ and $\bigcap\limits_{Y \in \mathcal{A}} Y$?
Relation between inf and intersection in a complete lattice
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combinatorics
discrete-mathematics
order-theory
lattice-orders
1 Answers
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In your lattice the order is given by $\subseteq$-relation so $A\subseteq B$ iff $a\in A\Rightarrow a\in B $ therefore $\inf_{(L.\subseteq)}(\mathcal{A}):=\bigcap\limits_{Y\in\mathcal{A}}Y$.
More generally if $(L,\leq)$ is a complete lattice then we say a $S \subseteq L$ is a closure system if for any $X\subseteq S$, we have $\inf_{(L,\leq)}(X)\in S$; and an easy result to prove is that every complete lattice is isomorphic to a closure system on the power set of some set.