Group-Valued Multisets
Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group of all integers. In this article, we introduce a significant advancement in hybrid sets through the concept of <i>group-valued multisets</i>. These multisets map elem...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/1/31 |
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| Summary: | Hybrid sets are defined as multisets having also negative multiplicities, i.e. as functions from a crisp set to the group of all integers. In this article, we introduce a significant advancement in hybrid sets through the concept of <i>group-valued multisets</i>. These multisets map elements of a set <i>X</i> to an arbitrary group, ensuring that each multiplicity has an inverse. This framework allows us to explore deeper relationships and correlations among the multiplicities of the elements within <i>X</i>. By involving the finitely supported sets, we study the new defined group-valued multisets over infinite universes of discourse in a finitary manner. After presenting the algebraic groups in the framework of finitely supported sets, we study the finitely supported group-valued multisets. We provide a finitary characterization of group-valued multisets over infinite universes of discourse, and obtain new results that generalize the properties of hybrid sets obtained in the Zermelo–Fraenkel framework. |
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| ISSN: | 2075-1680 |