A topological approach to the study of fuzzy measures.

*(English)*Zbl 0916.28015
Abramovich, Yuri (ed.) et al., Functional analysis and economic theory. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, dedicated to Charalambos Aliprantis on the occasion of his 50th birthday. Berlin: Springer. 17-46 (1998).

The authors work with an interesting structure \((P,\leq,+,\delta)\) (so-called \(\delta\)-\(l\)-semigroup), where \((P,+)\) is a partial order on \(P\) and \(\delta\) is a partial binary operation (\(x\delta y\) is defined if and only if \(x\leq y\)) satisfying some axioms [see H. Weber, Lect. Notes Math. 1089, 181-198 (1984; Zbl 0552.28011)]. MV algebra is a special case of \(\delta\)-\(l\)-semigroup. The authors prove an extension theorem for group-valued measures on \(\delta\)-\(l\)-semigroups. The second main result is an exhausting measure decomposition theorem. Using these general results, they obtain a complete description of all totally disconnected and all compact MV algebras.

For the entire collection see [Zbl 0893.00033].

For the entire collection see [Zbl 0893.00033].

Reviewer: B.Riečan (Bratislava)

##### MSC:

28E10 | Fuzzy measure theory |

##### Keywords:

fuzzy measures; group-valued measures; \(\delta\)-\(l\)-semigroups; exhausting measure decomposition theorem; MV algebras
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\textit{G. Barbieri} and \textit{H. Weber}, in: Functional analysis and economic theory. Based on the special session of the conference on nonlinear analysis and its applications in engineering and economics, Samos, Greece, July 1996, dedicated to Charalambos Aliprantis on the occasion of his 50th birthday. Berlin: Springer. 17--46 (1998; Zbl 0916.28015)