Let $X$ be a topological space.
$(a)$ $\text{BP}(X)$ denote σ-algebra of subsets of $X$ with the Baire Property $\text{BP}$.
$(b)$ $\text{MGR}(X)$ denote the σ-ideal of meager sets in $X$.
Let $[A]=\{B:B=*A\}$ be the$=^*$-equivalence class of $A$, and $\text{BP}(X)/\text{MGR}(X)$ be the quotient space $\{[A]:A \in \text{BP}(X)\}$.
Equip the quotient space $\text{BP}(X)/\text{MGR}(X)$ with the parcial ordering :
$[A]\leq [B] \Leftrightarrow A\setminus B\in \text{MGR}(X)$.
Question:
$(i)$ Is a Booelan σ-algebra?.
$(ii)$ If $X$ is a Baire space, is complete Booelan algebra?.
Any suggestion Thanks.