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Aspects of an internal logic for persistence
The foundational character of certain algebraic structures as Boolean
algebras and Heyting algebras is rooted in their potential to model classical
and constructive logic, respectively. In this paper we discuss the
contributions of algebraic logic to the study of persistence based on a new
operation on the ordered structure of the input diagram of vector spaces and
linear maps given by a filtration. Within the context of persistence theory, we
give an analysis of the underlying algebra, derive universal properties and
discuss new applications. We highlight the definition of the implication
operation within this construction, as well as interpret its meaning within
persistent homology, multidimensional persistence and zig-zag persistence