A combinatorial approach to prove an explicit form of some iteration groups Publications, to be read The m-Clean Graph of a Finite Commutative Ring Remarks on some generalized Sincov equations

Remarks on some generalized Sincov equations

Jointly written with JENS SCHWAIGER.

Aequationes mathematicae 99(6), 2745-2756 (2025). DOI 10.1007/s00010-025-01205-1.

Abstract. The Sincov equation f(x,z)=f(x,y)+f(y,z) has a long history. An excellent source is Gronau(2014). Under usual circumstances the general solution is given by f(x,y)=g(y)-g(x) with arbitrary g. This is also true, when the equation is satisfied for all x≤ y≤ z in a linearly ordered domain and for abelian groups as co-domain. In Piatek(2005) a result in this context is presented in the case that the domain is (only) partially ordered. We present a counter example and suggest positive results under mild additional hypotheses. In Bögel-Tasche(1974) and much better in chap. 7, The Lebesgue-Stieltes Integral of McShane(1944) the notion of additive interval functions is introduced. It seems that it went unnoticed till now that there is an intimate connection to the Sincov equation. This will be discussed in detail here.

harald.fripertinger "at" uni-graz.at, January 12, 2026

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