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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MDPI AG
2025-01-01
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| Online Access: | https://www.mdpi.com/2075-1680/14/1/31 |
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| author | Andrei Alexandru Gabriel Ciobanu |
| author_facet | Andrei Alexandru Gabriel Ciobanu |
| author_sort | Andrei Alexandru |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-a702022cca2f42f1aaa0896e99186a93 |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-a702022cca2f42f1aaa0896e99186a932025-01-24T13:22:12ZengMDPI AGAxioms2075-16802025-01-011413110.3390/axioms14010031Group-Valued MultisetsAndrei Alexandru0Gabriel Ciobanu1Institute of Computer Science, Romanian Academy, 700505 Iaşi, RomaniaInstitute of Computer Science, Romanian Academy, 700505 Iaşi, RomaniaHybrid 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.https://www.mdpi.com/2075-1680/14/1/31hybrid setsfinitely supported setsfinitely supported groupsgroup-valued multisetsDedekind finite groups |
| spellingShingle | Andrei Alexandru Gabriel Ciobanu Group-Valued Multisets Axioms hybrid sets finitely supported sets finitely supported groups group-valued multisets Dedekind finite groups |
| title | Group-Valued Multisets |
| title_full | Group-Valued Multisets |
| title_fullStr | Group-Valued Multisets |
| title_full_unstemmed | Group-Valued Multisets |
| title_short | Group-Valued Multisets |
| title_sort | group valued multisets |
| topic | hybrid sets finitely supported sets finitely supported groups group-valued multisets Dedekind finite groups |
| url | https://www.mdpi.com/2075-1680/14/1/31 |
| work_keys_str_mv | AT andreialexandru groupvaluedmultisets AT gabrielciobanu groupvaluedmultisets |