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Research and Publications - Department of Mathematics

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Browsing Research and Publications - Department of Mathematics by Author "Hanyang University"

Now showing 1 - 9 of 9
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    Fibonacci sequences in groupoids
    (Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama Tuscaloosa
    In this article, we consider several properties of Fibonacci sequences in arbitrary groupoids (i.e., binary systems). Such sequences can be defined in a left-hand way and a right-hand way. Thus, it becomes a question of interest to decide when these two ways are equivalent, i.e., when they produce the same sequence for the same inputs. The problem has a simple solution when the groupoid is flexible. The Fibonacci sequences for several groupoids and for the class of groups as special cases are also discussed. 2000 Mathematics Subject Classification: 20N02; 11B39.
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    Fuzzy rank functions in the set of all binary systems
    (Springer, 2016) Kim, Hee Sik; Neggers, J.; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
    In this paper, we introduce fuzzy rank functions for groupoids, and we investigate their roles in the semigroup of binary systems by using the notions of right parallelisms and rho-shrinking groupoids.
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    Generalized Fibonacci sequences in groupoids
    (Springer, 2013) Kim, Hee Sik; Neggers, J.; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
    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.
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    The Interaction between Fuzzy Subsets and Groupoids
    (Hindawi, 2014) Shin, Seung Joon; Kim, Hee Sik; Neggers, J.; University of Michigan; Hanyang University; University of Alabama Tuscaloosa
    We discuss properties of a class of real-valued functions on a set X-2 constructed as finite (real) linear combinations of functions denoted as [(X, *); mu], where (X, *) is a groupoid (binary system) and mu is a fuzzy subset of X and where [(X., *); mu] (x, y) := mu (x * y) - min {mu(x), mu(y)}. Many properties, for example, mu being a fuzzy subgroupoid of (X, *), can be restated as some properties of [(X, *); mu]. Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin (X); square) for example.
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    (n-1)-Step Derivations on n-Groupoids: The Case n=3
    (Hindawi, 2014) Alshehri, N. O.; Kim, Hee Sik; Neggers, J.; King Abdulaziz University; Hanyang University; University of Alabama Tuscaloosa
    We define a ranked trigroupoid as a natural followup on the idea of a ranked bigroupoid. We consider the idea of a derivation on such a trigroupoid as representing a two-step process on a pair of ranked bigroupoids where the mapping d is a self-derivation at each step. Following up on this idea we obtain several results and conclusions of interest. We also discuss the notion of a couplet (D, d) on X, consisting of a two-step derivation d and its square D = d circle d, for example, whose defining property leads to further observations on the underlying ranked trigroupoids also.
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    On Abelian and Related Fuzzy Subsets of Groupoids
    (Hindawi, 2013) Shin, Seung Joon; Kim, Hee Sik; Neggers, J.; University of Michigan; Hanyang University; University of Alabama Tuscaloosa
    We introduce the notion of abelian fuzzy subsets on a groupoid, and we observe a variety of consequences which follow. New notions include, among others, diagonal symmetric relations, several types of quasi orders, convex sets, and fuzzy centers, some of whose properties are also investigated.
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    On Fibonacci functions with Fibonacci numbers
    (Springer, 2012) Han, Jeong Soon; Kim, Hee Sik; Neggers, Joseph; Hanyang University; University of Alabama Tuscaloosa
    In this paper we consider Fibonacci functions on the real numbers R, i.e., functions f : R -> R such that for all x is an element of R, f(x + 2) = f(x + 1) + f(x). We develop the notion of Fibonacci functions using the concept of f f-even and f-odd functions. Moreover, we show that if f is a Fibonacci function then lim(x ->infinity) f(x+1)/f(x) = 1+root 5/2.
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    On Fibonacci functions with periodicity
    (Springer, 2012) Kim, Hee Sik; Neggers, Joseph; So, Keum Sook; Hanyang University; University of Alabama Tuscaloosa; Hallym University
    In this paper we discuss Fibonacci functions using the (ultimately) periodicity and we also discuss the exponential Fibonacci functions. Especially, given a non-negative real-valued function, we obtain several exponential Fibonacci functions.
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    Several types of groupoids induced by two-variable functions
    (Springer, 2016) Allen, P. J.; Kim, Hee Sik; Neggers, J.; University of Alabama Tuscaloosa; Hanyang University
    In this paper, we introduce the concept of several types of groupoids related to semigroups, viz., twisted semigroups for which twisted versions of the associative law hold. Thus, if (X, *) is a groupoid and if phi : X-2 -> X-2 is a function phi (a, b) = (u, v), then (X, *) is a left-twisted semigroup with respect to phi if for all a, b, c is an element of X, a * (b * c) = (u * v) * c. Other types are right-twisted, middle-twisted and their duals, a dual left-twisted semigroup obeying the rule (a * b) * c = u * (v * c) for all a, b, c is an element of X. Besides a number of examples and a discussion of homomorphisms, a class of groupoids of interest is the class of groupoids defined over a field (X,+, .) via a formula x * y = lambda x + mu y, with lambda, mu is an element of X, fixed structure constants. Properties of these groupoids as twisted semigroups are discussed with several results of interest obtained, e.g., that in this setting simultaneous left-twistedness and right-twistedness of (X, *) implies the fact that (X, *) is a semigroup.