Set-Operational Properties of Semiatoms in Non-additive Measure Theory

The semiatom is a basic concept in the non-additive measure theory, or the fuzzy measure theory, and has been used for applications of the theory (T. Murofushi et al., 1997, Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 5, 563–585; and T. Murofushi and M. Sugeno, 2000, ibid. 8, 385–415). This paper shows several properties of semiatoms on set operations:union, intersection, difference, symmetric difference, countable union, and countable intersection. Characteristic consequences are as follows:if S and T are semiatoms, and if S ∩ T is non-null, then S ∪ T and S ∩ T are semiatoms; moreover, if S\T and T\S are non-null, then S\T, T\S, S T also are semiatoms