Large Sets in Boolean and Non-Boolean Groups and Topology

Polycyclic, metabelian or soluble of type (FP)$_{\infty}$ groups with boolean algebra of rational sets and biautomatic soluble groups are virtually abelian

Let $G$ be a polycyclic, metabelian or soluble of type (FP)$_{\infty}$ group such that the class $Rat(G)$ of all rational subsets of $G$ is a boolean algebra. Then $G$ is virtually abelian. Every soluble biautomatic group is virtually abelian.

全不连通半群紧化：闭开集完全布尔代数的非内省性

Totally Disconnected Semigroup Compactifications: Non-Introversion of the Full Boolean Algebra of Clopen Sets

通过单个闭开集和两个相关网刻画拓扑群G的闭开集完全布尔代数B(G)左内省的条件，证明第一可数σ-紧全不连通局部紧群G中B(G)左内省当且仅当G紧或离散，肯定回答Stephens和Stokke的问题。

Boolean elements in the Bruhat order

We show that $w\in W$ is boolean if and only if it avoids a set of Billey-Postnikov patterns, which we describe explicitly. Our proof is based on an analysis of inversion sets, and it is in large part type-uniform. We also introduce the notion of linear pattern avoidance, and show that boolean elements are characterized by avoiding just the $3$ linear patterns $s_1 s_2 s_1 \in W(A_2)$, $s_2 s_1 s_3 s_2 \in W(A_3)$, and $s_2 s_1 s_3 s_4 s_2 \in W(D_4)$. We also consider the more general case of $k$-boolean Weyl group elements. We say that $w\in W$ is $k$-boolean if every reduced expression for …