Density of type-dependent sets in Krull monoids with analytic structure

We describe structural and quantitative properties of type-dependent sets in monoids with suitable analytic structure, including simple analytic monoids, introduced by Kaczorowski (Semigroup Forum 94:532–555, 2017. 10.1007/s00233-016-9778-9), and formations, as defined by Geroldinger and Halter-Koch (Non-unique factorizations, Chapman and Hall, Boca Raton, 2006. 10.1201/9781420003208). We propose the notions of rank and degree to measure the size of a type-dependent set in structural terms. We also consider various notions of regularity of type-dependent sets, related to the analytic properties of their zeta functions, and obtain results on the counting functions of these sets.

An algebraic characterisation of ample type I groupoids

We give algebraic characterisations of the type I and CCR properties for locally compact second countable, ample Hausdorff groupoids in terms of subquotients of its Boolean inverse semigroup of compact open local bisections. It yields in turn algebraic characterisations of both properties for inverse semigroups with meets in terms of subquotients of their Booleanisation.