On Self-Aggregations of Min-Subgroups
Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-sub...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/3/201 |
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| author | Carlos Bejines Sergio Ardanza-Trevijano Jorge Elorza |
| author_facet | Carlos Bejines Sergio Ardanza-Trevijano Jorge Elorza |
| author_sort | Carlos Bejines |
| collection | DOAJ |
| description | Preservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets. |
| first_indexed | 2024-03-10T07:52:59Z |
| format | Article |
| id | doaj.art-28aa926760664b2dbddd54f377717f0b |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T07:52:59Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-28aa926760664b2dbddd54f377717f0b2023-11-22T12:02:53ZengMDPI AGAxioms2075-16802021-08-0110320110.3390/axioms10030201On Self-Aggregations of Min-SubgroupsCarlos Bejines0Sergio Ardanza-Trevijano1Jorge Elorza2Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, 121 35 Prague, Czech RepublicDepartamento de Física y Matemática Aplicada, University of Navarra, 31080 Pamplona, SpainDepartamento de Física y Matemática Aplicada, University of Navarra, 31080 Pamplona, SpainPreservation of structures under aggregation functions is an active area of research with applications in many fields. Among such structures, min-subgroups play an important role, for instance, in mathematical morphology, where they can be used to model translation invariance. Aggregation of min-subgroups has only been studied for binary aggregation functions. However, results concerning preservation of the min-subgroup structure under binary aggregations do not generalize to aggregation functions with arbitrary input size since they are not associative. In this article, we prove that arbitrary self-aggregation functions preserve the min-subgroup structure. Moreover, we show that whenever the aggregation function is strictly increasing on its diagonal, a min-subgroup and its self-aggregation have the same level sets.https://www.mdpi.com/2075-1680/10/3/201aggregation functionT-subgroupstrictly monotone function |
| spellingShingle | Carlos Bejines Sergio Ardanza-Trevijano Jorge Elorza On Self-Aggregations of Min-Subgroups Axioms aggregation function T-subgroup strictly monotone function |
| title | On Self-Aggregations of Min-Subgroups |
| title_full | On Self-Aggregations of Min-Subgroups |
| title_fullStr | On Self-Aggregations of Min-Subgroups |
| title_full_unstemmed | On Self-Aggregations of Min-Subgroups |
| title_short | On Self-Aggregations of Min-Subgroups |
| title_sort | on self aggregations of min subgroups |
| topic | aggregation function T-subgroup strictly monotone function |
| url | https://www.mdpi.com/2075-1680/10/3/201 |
| work_keys_str_mv | AT carlosbejines onselfaggregationsofminsubgroups AT sergioardanzatrevijano onselfaggregationsofminsubgroups AT jorgeelorza onselfaggregationsofminsubgroups |