<p>The <i>Sincov</i> equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(f(x,z)=f(x,y)+f(y,z)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> has a long history. An excellent source is Gronau(2014). Under usual circumstances the general solution is given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(f(x,y)=g(y)-g(x)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>-</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> with arbitrary <i>g</i>. This is also true, when the equation is satisfied for all <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(x\le y\le z\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>≤</mo> <mi>y</mi> <mo>≤</mo> <mi>z</mi> </mrow> </math></EquationSource> </InlineEquation> in a linearly ordered domain and for abelian groups as co-domain. In Pia̧tek(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.&#xa0;7, The Lebesgue-Stieltes Integral of McShane(1944) the notion of <i>additive interval</i> 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.</p>