Variations on Effect Algebras

Abstract

The aim of the present paper is to introduce and investigate the variations of (non-additive) functions defined on effect algebras. The notion of the variation of a general function is introduced on an effect algebra \(L\) and it is proved that it always exists, but in general case it is not unique; the notions of orthogonal variation \(\overline{m},\) chain variation \(|m|\) and inclusion variation \(|m|_i\) of a real-valued function \(m\) defined on \(L\) are introduced and its properties are discussed elaborately. Finally, it is also proved that the orthogonal variation \(\overline{m}\) of a modular measure \(m\) defined on a \(\sigma \)-complete \(D\)-lattice \(L\) is the smallest variation on \(L\).