Pseudo-atoms, atoms and a Jordan type decomposition in effect algebras |
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Authors: | Mona Khare Akhilesh Kumar Singh |
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Affiliation: | a Department of Mathematics, University of Allahabad, Allahabad 211 002, India b Allahabad Mathematical Society, 10 C.S.P. Singh Marg, Allahabad 211 001, India |
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Abstract: | 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. |
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Keywords: | Effect algebra Pseudo-atom Atom Null-additivity Null-null-additivity Total variation Jordan type decomposition |
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