Universal Fibonacci sequences and UFS-groupoids
Fuente:
arXiv
Guardado en:
| Autor principal: | |
|---|---|
| Formato: | Preprint |
| Publicado: |
2026
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866913092930109440 |
|---|---|
| author | Klimov, Petr |
| author_facet | Klimov, Petr |
| contents | In a binary groupoid $(G, *)$, a Fibonacci sequence is a recurrent sequence defined by $f_1 = a, f_2 = b, \ldots, f_n = f_{n - 2} * f_{n - 1}$. A universal Fibonacci sequence (UFS) is a singly or doubly infinite sequence whose set of suffixes coincides precisely with the set of all Fibonacci sequences in the groupoid.
This paper studies UFS-groupoids, i.e., groupoids that admit a universal Fibonacci sequence. It is shown that every nontrivial UFS-groupoid is at most countable, locally cyclic, and non-power-associative; that the right cancellation property and the right quasigroup property hold for all pairs of elements except possibly one and two, respectively; that no neutral element or zero element exists; and that there is at most one idempotent element. It is proved that any UFS-groupoid whose universal Fibonacci sequence is not doubly infinite strictly preperiodic is cyclic. It has also been proved that the class of UFS-groupoids is closed under taking subgroupoids and homomorphic images, but is not closed under finite direct products. The structure of subgroupoids of UFS-groupoids is described.
A complete classification of UFS-groupoids is given in terms of the cardinality of $G$ and the periodicity of the universal Fibonacci sequences. Finite UFS-groupoids are described combinatorially via de Bruijn sequences. The number of distinct UFS-groupoids on a finite set is determined, and explicit constructions are provided for both finite and infinite cases across all periodicity classes, including embeddings of UFS-groupoids as subgroupoids into other UFS-groupoids and infinitely generated UFS-groupoids. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_05754 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal Fibonacci sequences and UFS-groupoids Klimov, Petr Group Theory Combinatorics Rings and Algebras In a binary groupoid $(G, *)$, a Fibonacci sequence is a recurrent sequence defined by $f_1 = a, f_2 = b, \ldots, f_n = f_{n - 2} * f_{n - 1}$. A universal Fibonacci sequence (UFS) is a singly or doubly infinite sequence whose set of suffixes coincides precisely with the set of all Fibonacci sequences in the groupoid. This paper studies UFS-groupoids, i.e., groupoids that admit a universal Fibonacci sequence. It is shown that every nontrivial UFS-groupoid is at most countable, locally cyclic, and non-power-associative; that the right cancellation property and the right quasigroup property hold for all pairs of elements except possibly one and two, respectively; that no neutral element or zero element exists; and that there is at most one idempotent element. It is proved that any UFS-groupoid whose universal Fibonacci sequence is not doubly infinite strictly preperiodic is cyclic. It has also been proved that the class of UFS-groupoids is closed under taking subgroupoids and homomorphic images, but is not closed under finite direct products. The structure of subgroupoids of UFS-groupoids is described. A complete classification of UFS-groupoids is given in terms of the cardinality of $G$ and the periodicity of the universal Fibonacci sequences. Finite UFS-groupoids are described combinatorially via de Bruijn sequences. The number of distinct UFS-groupoids on a finite set is determined, and explicit constructions are provided for both finite and infinite cases across all periodicity classes, including embeddings of UFS-groupoids as subgroupoids into other UFS-groupoids and infinitely generated UFS-groupoids. |
| title | Universal Fibonacci sequences and UFS-groupoids |
| topic | Group Theory Combinatorics Rings and Algebras |
| url | https://arxiv.org/abs/2604.05754 |