Lattice valuations, medians and majorities (Q686486)

The paper concerns semilattices and lattices. On a semilattice an upper valuation or a lower valuation, on a lattice a valuation is introduced. A valuation on a finite lattice \(L\) is a strongly isotone mapping of \(L\) into the set of real numbers which satisfies \(v(x\vee y)+ v(x\wedge y)= v(x)+ v(y)\) for any two elements \(x\), \(y\) of \(L\). A valuation \(v\) on \(L\) induces a distance \(d_ v\) on \(L\) given by \(d_ v(x,y)= 2v(x\vee y)- v(x)- v(y)\) and similarly it is defined for an upper valuation on a semilattice. It can be studied as a distance on a graph. An arbitrary \(p\)-tuple \(\pi=(x_ 1,\dots, x_ n)\) of elements of \(L\) is called a profile. A remoteness \(r_ \pi\) of an element \(x\) with respect to \(\pi\) is defined so that \(r_ \pi(x)= \sum_{i=1}^ p d_ v(x,x_ i)\). The element \(x\) for which \(r_ \pi(x)\) is minimum is called a \(\pi\)-median. Some theorems on these concepts are proved.