Integer-valued valuations
Andrii Ilienko, Ilya Molchanov, Tommaso Vison\`a

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