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This is a follow-up to my previous question, which has a negative answer. Here is the most general version that I'm interested: Does there exist a Boolean algebra $A$, an infinite cardinal $\kappa$, ...

Define an operator $\mathop{\vec{\bigcup}}$ as follows: Definition. Whenever $A$ is an $I$-indexed family of sets, where $I$ is a totally-ordered set, we have $$\mathop{\vec{\bigcup}}_{i \in I} A_i ...

Let $(X,\tau)$ be a topological space. $A\subseteq X$ is said to be regular open if $A = \text{int}(\text{cl}(A))$ and let $\text{RO}(X,\tau)$ denote the collection of regular open sets of $X$. A ...

Let $(P,\leq)$ be a poset. The interval topology $\tau_i(P)$ on $P$ is generated by $$\{P\setminus\downarrow x : x\in P\} \cup \{P\setminus\uparrow x : x\in P\},$$ where $\downarrow x = \{y\in P: y\...

Let A and B be Boolean algebras and $h:A\rightarrow B$ a Boolean embedding. If every element of $B$ can be expressed both as a join of meets and as a meet of joins of elements in $h(A)$, then the ...