Generalized Fibonacci sequences in groupoids
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RESEARCH
Open Access
Generalized Fibonacci sequences in groupoids Hee Sik Kim1 , J Neggers2 and Keum Sook So3* *
Correspondence: [email protected] Department of Mathematics, Hallym University, Chuncheon, 200-702, Korea Full list of author information is available at the end of the article 3
Abstract In this paper, we introduce the notion of generalized Fibonacci sequences over a groupoid and discuss it in particular for the case where the groupoid contains idempotents and pre-idempotents. Using the notion of Smarandache-type P-algebra, we obtain several relations on groupoids which are derived from generalized Fibonacci sequences. MSC: 11B39; 20N02; 06F35 Keywords: (generalized) Fibonacci sequences; (L4 , LRL2 , R4 )-groupoids; d/BCK-algebra; (pre-)idempotent; Smarandache disjoint
1 Introduction Fibonacci-numbers have been studied in many different forms for centuries and the literature on the subject is consequently incredibly vast. Surveys and connections of the type just mentioned are provided in [] and [] for a very minimal set of examples of such texts, while in [] an application (observation) concerns itself with the theory of a particular class of means which has apparently not been studied in the fashion done there by two of the authors of the present paper. Han et al. [] studied a Fibonacci norm of positive integers, and they presented several conjectures and observations. Given the usual Fibonacci-sequences [, ] and other sequences of this type, one is naturally interested in considering what may happen in more general circumstances. Thus, one may consider what happens if one replaces the (positive) integers by the modulo integer n or what happens in even more general circumstances. The most general circumstance we will deal with in this paper is the situation where (X, ∗) is actually a groupoid, i.e., the product operation ∗ is a binary operation, where we assume no restrictions a priori. Han et al. [] considered several properties of Fibonacci sequences in arbitrary groupoids. The notion of BCK -algebras was introduced by Iséki and Imai in . This notion originated from both set theory and classical and non-classical propositional calculi. The operation ∗ in BCK -algebras is an analogue of the set-theoretical difference. Nowadays, BCK -algebras have been studied by many authors and they have been applied to many branches of mathematics such as group theory, functional analysis, probability theory, topology and fuzzy theory [–] and so on. We refer to [, ] for further information on BCK/BCI-algebras. Let (X, ∗) be a groupoid (or an algebra). Then (X, ∗) is a Smarandache-type P-algebra if it contains a subalgebra (Y , ∗), where Y is non-trivial, i.e., |Y | ≥ , or Y contains at © 2013 Kim et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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