Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras

The aim of this paper is to introduce and investigate the concept of pseudo-atoms of a real-valued function m defined on an effect algebra L; a few examples of pseudo-atoms and atoms are given in the context of null-additive, null–null-additive and pseudo-null-additive functions and also, some fundamental results for pseudo-atoms under the assumption of null–null-additivity are established. The notions of total variation |m|, positive variation m+ and negative variation m− of a real-valued function m on L are studied elaborately and it is proved for a modular measure m (which is of bounded total variation) defined on a D-lattice L that, m is pseudo-atomic (or atomic) if and only if its total variation |m| is pseudo-atomic (or atomic). Finally, a Jordan type decomposition theorem for an extended real-valued function m of bounded total variation defined on an effect algebra L is proved and some properties on decomposed parts of m such as continuity from below, pseudo-atomicity (or atomicity) and being measure, are discussed. A characterization for the function m to be of bounded total variation is established here and used in proving above-mentioned Jordan type decomposition theorem.