Revisiting fuzzy multisets: clarifying ambiguities and the underlying algebraic structure
We revisit the theory of fuzzy multisets to address formal inconsistencies in existing definitions. Building on Miyamoto’s intuitive membership sequences, we propose two revised frameworks on a common foundation. The first one (Option 1) retains explicit multiplicities even at null memberships, ther...
| Autores: | , |
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| Tipo de documento: | artigo |
| Estado: | Versão publicada |
| Data de publicação: | 2026 |
| País: | España |
| Recursos: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositório: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:dnet:digitalcsic_::a6bb35377a82c35598b64a93bce45554 |
| Acesso em linha: | http://hdl.handle.net/10261/431740 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Multiset Fuzzy multiset De Morgan algebra n-dimensional fuzzy sets |
| Resumo: | We revisit the theory of fuzzy multisets to address formal inconsistencies in existing definitions. Building on Miyamoto’s intuitive membership sequences, we propose two revised frameworks on a common foundation. The first one (Option 1) retains explicit multiplicities even at null memberships, thereby distinguishing clearly between “no assignment” and “zero assignment”, while the second one (Option 2) restricts attention to the subfamily of fuzzy multisets that uniformly enforce zero multiplicity to null memberships. Under both frameworks, the new operations of union, intersection, and inclusion satisfy the usual algebraic laws, most notably the absorption law, which fails in the original treatment, and endow the family of fuzzy multisets with a lattice structure. Moreover, within Option 2 we define a complement operation by invoking a top multiset; this complement is order-reversing and involutive, thereby equipping bounded fuzzy multisets with a De Morgan algebra structure. Finally, we prove that n-dimensional fuzzy sets are isomorphic to n-bounded fuzzy multisets, and we study the formal connections between the lattices of general fuzzy multisets and multidimensional fuzzy sets. |
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