Communications of the
Korean Mathematical Society CKMS

In this paper, we introduce an operation denoted by $[Br_e]$, a bracket operation, which maps an arbitrary groupoid $(X,*)$ on a set $X$ to another groupoid $(X,\bullet)=[Br_e](X,*)$ which on groups corresponds to sending a pair of elements $(x,y)$ of $X$ to its commutator $xyx^{-1}y^{-1}$. When applied to classes such as $d$-algebras, $BCK$-algebras, a variety of results is obtained indicating that this construction is more generally useful than merely for groups where it is of fundamental importance.

Keywords: bracket function, $e$-bracket image algebra, $e$-bracket-abelian, $d/B/BCK$-algebra, Smarandache disjoint, $Bin(X)$

MSC numbers: Primary 20N02, 08A02, 06F35