We introduce $X$-to-join representations of 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-Sz\'ep product right LCM semigroup by Brownlowe, Ramagge, Robertson and Whittaker.
Categories: math.RA math.OA