arXiv subjects

Andrew B. Apps

Publications and source records attributed to Andrew B. Apps.

2 recordsLinked to original sources

arXiv2025

Invariants for metrisable locally compact Boolean spaces

Pierce identified 3 invariants of a compact metrisable Boolean space, derived from its Cantor-Bendixson sequence, that determine the space up to homeomorphism. For locally compact spaces we define an additional invariant, the compact rank, and show that these 4 invariants determine a locally compact metrisable Boolean space up to homeomorphism. We also identify which combinations of the 4 invariants can arise in practice. A Boolean ring and its associated Boolean space are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Spaces in this important sub-class of Boolean spaces can be well described (uniquely in the case of compact spaces) by an extended PO system (poset with a distinguished subset). We define the Cantor-Bendixson sequence and associated invariants for a PO system, and show that almost all of the invariant information for a primitive space can be recovered from that of an associated extended PO system. We also show how the primitivity of a Boolean space corresponds to a notion of primitivity of the additive measure associated with the rank function of a space, which in turn depends on the additive measure being sufficiently 'self-similar'. We use these ideas to develop a method for constructing non-primitive spaces.

math.LO↗

arXiv2023

Partitions of primitive Boolean spaces

A Boolean ring and its Stone space (Boolean space) are primitive if the ring is disjointly generated by its pseudo-indecomposable (PI) elements. Hanf showed that a primitive PI Boolean algebra can be uniquely defined by a structure diagram. In a previous paper we defined trim $P$-partitions of a Stone space, where $P$ is a PO system (poset with a distinguished subset), and showed how they provide a physical representation within the Stone space of these structure diagrams. In this paper we study the class of trim partitions of a fixed primitive Boolean space, which may not be compact, and show how they can be structured as a quasi-ordered set via an appropriate refinement relation. This refinement relation corresponds to a surjective morphism of the associated PO systems, and we establish a quasi-order isomorphism between the class of well-behaved partitions of a primitive space and a class of extended PO systems. We also define rank partitions, which generalise the rank diagrams introduced by Myers, and the ideal completion of a trim $P$-partition, whose underlying PO system is the ideal completion of $P$, and show that rank partitions are just the ideal completions of trim partitions. In the process, we extend a number of existing results regarding primitive Boolean algebras or compact primitive Boolean spaces to locally compact Boolean spaces.

math.LO↗