Universal Fibonacci sequences and UFS-groupoids

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.