Boolean inverse semigroups

Abstract: Boolean inverse semigroups are inverse semigroups whose idempotents admit a structure of a Boolean algebra and possessing joins of compatible pairs of elements. Non-commutative Stone duality connects Boolean inverse semigroups with Stone groupoids which are \'etale topological groupoids whose space of identities is a Stone space. The focus of the talk will be on the speaker's recent work on $X$-to-join representations of inverse semigroups in Boolean inverse semigroups which are a relaxation of the notion of a cover-to-join representation. We construct the universal $X$-to-join Booleanization of an inverse semigroup $S$ as a weakly meet-preserving quotient of the universal Booleanization ${\mathrm B}(S)$ and show that all such quotients of ${\mathrm B}(S)$ arise via $X$-to-join representaions. As an application, we provide groupoid models for the intermediate boundary quotients of the $C^*$-algebra of a Zappa-Szép product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.

category theorygroup theoryrings and algebras

Audience: researchers in the topic

York semigroup seminar

Series comments: Description: Semigroup-related research talks at University of York.

Email Nora Szakacs at nora.szakacs@york.ac.uk for the meeting password.