Interval Replacements of Persistence Modules

We define (1) a notion of a compression system ξ for a finite poset P, which assigns each interval subposet I to a poset morphism ξI QI → P and (2) an I-rank of a persistence module M with respect to ξ, the family of which is called the interval rank invariant. A compression system ξ makes it possible to define the interval replacement (also called the interval-decomposable approximation) not only for 2D persistence modules but also for any persistence modules over any finite poset. We will show that the forming of the interval replacement preserves the interval rank invariant that is a stronger property than the preservation of the usual rank invariant. Moreover, we will give an explicit formula of the I-rank of M with respect to ξ in terms of the structure linear maps of M under a mild existence condition of joins and meets in I in the case where ξI is the inclusion of I into P, or more generally, ξI ''essentially covers'' I.