A metric graph is a geometric realization of a finite graph by identifying each edge with a real interval. A divisor on a metric graph Γ is an element of the free abelian group on Γ. The rank of a divisor on a metric graph is a concept appearing in the RiemannRoch theorem for metric graphs (or tropical curves) due to Gathmann and Kerber [7], and Mikhalkin and Zharkov [10]. We define a rank-dete...

We study divisors in a complex manifold in view of the property that the algebra of logarithmic differential operators along the divisor is generated by logarithmic vector fields. We give • a sufficient criterion for the property, • a simple proof of F.J. Calderón–Moreno’s theorem that free divisors have the property, • a proof that divisors in dimension 3 with only isolated quasi–homogeneous s...

Some conditions equivalent to a strong quasi-divisor property (SQDP) for a partly ordered group G are derived. It is proved that if G is defined by a family of t-valuations of finite character, then G admits an SQDP if and only if it admits a quasi-divisor property and any finitely generated t-ideal is generated by two elements. A topological density condition in topological group of finitely g...