Integer-valued valuations

TL;DR

This paper characterizes integer-valued valuations on planar sets without invariance assumptions, explores the effects of relaxing certain conditions, and introduces a new product construction for these valuations.

Contribution

It provides a complete classification of planar monotone $\sigma$-continuous integer-valued valuations and introduces a novel product construction for such valuations.

Findings

Complete characterization of planar monotone $\sigma$-continuous integer-valued valuations

Full classification of line valuations when dropping monotonicity or $\sigma$-continuity

Introduction of a new product construction for these valuations

Abstract

We obtain a complete characterization of planar monotone $\sigma$-continuous valuations taking integer values, without assuming invariance under any group of transformations. We further investigate the consequences of dropping monotonicity or $\sigma$-continuity and give a full classification of line valuations. We also introduce a construction of the product for valuations of this type.