Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

Article

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Bull. Korean Math. Soc. 2017; 54(4): 1373-1386

Online first article June 2, 2017      Printed July 31, 2017

https://doi.org/10.4134/BKMS.b160593

Copyright © The Korean Mathematical Society.

Order related concepts for arbitrary groupoids

Hee Sik Kim, Joseph Neggers, and Keum Sook So

Hanyang University, University of Alabama, Hallym University

Abstract

In this paper, we introduce and explore suggested notions of `above', `below' and `between' in general groupoids, $Bin(X)$, as well as in more detail in several well-known classes of groupoids, including groups, semigroups, selective groupoids (digraphs), $d/BCK$-algebras, linear groupoids over fields and special cases, in order to illustrate the usefulness of these ideas. Additionally, for groupoid-classes (e.g., $BCK$-algebras) where these notions have already been accepted in a standard form, we look at connections between the several definitions which result from our introduction of these ideas as presented in this paper.

Keywords: below, above, between, $d/BCK$-algebra, poset, $Bin(X)$, order-preserving, $\beta$-medial

MSC numbers: Primary 20N02, 20M10, 06A06